L_1 embeddings of the Heisenberg group and fast estimation of graph isoperimetry

Pro ceedings of the Internat ional Congress of Mathematicians Hyderabad, India, 2010 L 1 em b edding s of the Hei s en b erg gr o up and fast esti mation of graph is o p erimetry Assaf Naor ∗ Abstract. W e survey connections b etw een the t heory of bi-Lipschitz embeddings and the Sparsest Cut Problem in combinatorial optimization. The story of th e Sparsest Cut Problem is a striking example of the deep interpla y b etw een analysis, geometry , and probabilit y on the one hand, an d computational issues in discrete mathematics on the other. W e explain how the key ideas evo lved ov er t he past 20 years , emphasizing the intera ctions with Banach space theory , geometric measure theory , and geometric group theory . As an imp ortant illustrativ e example, w e shall examine recently established connections to the the struct ure of the Heis enberg gro up, and the incompatibility of its Carnot-Carath ´ eodory geometry with the geometry of the Leb esgue space L 1 . Mathematics Sub ject Classification (2000). 46B85 , 30L05, 46B80, 51F99. Keywords. Bi-Lipschitz embeddings, S parsest Cut Problem, H eisenberg group. 1. In tro duction Among the common definitions of the Heisenberg group H , it will b e conv enien t for us to work he r e with H modeled a s R 3 , equipp ed with the group pro duct ( a, b, c ) · ( a ′ , b ′ , c ′ ) def = ( a + a ′ , b + b ′ , c + c ′ + ab ′ − ba ′ ). The integer lattice Z 3 is then a discrete co compact subgroup of H , denoted b y H ( Z ), which is generated by the finite symmetric set { ( ± 1 , 0 , 0) , (0 , ± 1 , 0) , (0 , 0 , ± 1) } . The word metric on H ( Z ) induced by this gene r ating s e t will b e denoted by d W . As no ted by Semmes [66], a differentiabilit y res ult of Pansu [61] implies that the metric space ( H ( Z ) , d W ) do es not admit a bi-Lipschitz embedding into R n for any n ∈ N . This was extended by Pauls [6 2] to bi-Lipschitz non-e m be dda bilit y results of ( H ( Z ) , d W ) into metric spaces with either low er o r upp er curv ature b ounds in the sense of Alexandrov. In [5 2, 27] it w as obser ved that Pansu’s differentiabil- it y argument extends to Banach space ta r gets with the Radon- Nikod´ ym pr op erty (see [14, Ch. 5]), and hence H ( Z ) do es not admit a bi-Lipschitz e mbedding into, say , a Banach space which is either r eflexive or is a sepa rable dual; in par ticular H ( Z ) do es not a dmit a bi- Lipschitz embedding into a ny L p ( µ ) space , 1 < p < ∞ , or into the sequence spa ce ℓ 1 . ∗ Researc h suppor ted in part b y NSF grants CCF- 0635078 and CCF-0832795, B SF grant 2006009, and the Pac k ard F oundation. 2 Assaf Naor The embeddability of H ( Z ) in to the function space L 1 ( µ ), when µ is non-atomic, turned out to be mu ch harder to s ettle. This question is of particular imp ortance since it is well under sto o d that for µ non-atomic, L 1 ( µ ) is a spa ce fo r whic h the dif- ferentiabilit y re s ults quo ted ab ov e manifestly break down. Nev ertheless, Cheeg er and Kleiner [2 6, 25] in tro duced a nov el notion of differentiability for which they could prove a differentiabilit y theorem for Lipschitz maps fro m the Heisenberg group to L 1 ( µ ), thus establishing that H ( Z ) do es not a dmit a bi- Lipschitz embed- ding in to any L 1 ( µ ) space . Another motiv ation for the L 1 ( µ ) embedda bility question for H ( Z ) orig inates from [52], where it was established that it is co nnec ted to the Sparses t Cut Pr oblem in the field of co m binatorial optimizatio n. F or this application it was of imp or - tance to obtain qua n titative estimates in the L 1 ( µ ) non-embedda bilit y results for H ( Z ). It turns out that establishing such estimates is quite subtle, a s they req uir e ov ercoming finitar y issues that do not a rise in the infinite s etting of [25, 28]. The following tw o theo rems were proved in [29, 30]. Both theorems follow painlessly from a more genera l theorem that is stated and discusse d in Sec tio n 5 .4. Theorem 1.1. Ther e exists a universal c ons t ant c > 0 such that any emb e dding into L 1 ( µ ) of t he r estriction of the wor d metric d W to the n × n × n grid { 1 , . . . , n } 3 incurs distortion & (log n ) c . F ollo wing Gr omov [38], the compre s sion rate o f f : H ( Z ) → L 1 ( µ ), denoted ω f ( · ), is defined as the la rgest non-decreasing function suc h that for all x, y ∈ H ( Z ) we hav e k f ( x ) − f ( y ) k 1 > ω f ( d W ( x, y )) (see [7] for mo r e information on this topic). Theorem 1.2. Ther e ex ists a universal c onstant c > 0 such that for every function f : H ( Z ) → L 1 ( µ ) which is 1 -Lipschitz with r esp e ct to the wor d metric d W , we have ω f ( t ) . t/ (log t ) c for al l t > 2 . Ev a luating the supremum of those c > 0 for which Theore m 1.1 holds true remains an imp ortant op en question, with ge o metric s ig nificance a s well a s imp or- tance to theoretical computer s c ience. Conceiv ably we could get c in Theore m 1.1 to b e arbitra rily c lose to 1 2 , which would b e sharp since the results of [8, 6 4] imply (see the explanation in [41]) that the metric spa ce  { 1 , . . . , n } 3 , d W  embeds into ℓ 1 with distortion . √ log n . Similar ly , we do not know the b est p os sible c in Theorem 1.2; 1 2 is aga in the limit here since it was shown in [69] that there exists a 1-Lipschitz mapping f : H ( Z ) → ℓ 1 for which ω f ( t ) & t/ ( √ log t · log lo g t ). The purp ose of this article is to de s crib e the ab ov e non-embeddability r esults fo r the Heisenber g gro up. Since one o f the motiv atio ns for thes e inv estigations is the application to the Sparsest Cut Problem, we a ls o include here a detailed discussion of this pro blem fro m theoretical co mputer sc ie nce, and its dee p connections to metric geo metr y . Our go al is to present the ideas in a wa y tha t is accessible to mathematicians who do not necess arily hav e background in computer scienc e . Ac kno wl edgements. I am gr ateful to the following p eople for helpful comments and suggestions on e arlier versions of this manuscript: Tim Austin, Keith Ball, Subhash Khot, Bruce K leiner, Russ Lyons, Manor Mendel, Gideon Schech tman, Lior Silb erman. Em b eddings of the H eisen b erg group 3 2. Em b eddings A metric space ( M , d M ) is said to embed with disto rtion D > 1 into a metric spa ce ( Y , d Y ) if there ex ists a mapping f : M → Y , a nd a scaling factor s > 0, such tha t for all x, y ∈ M we hav e sd M ( x, y ) 6 d Y ( f ( x ) , f ( y )) 6 D sd M ( x, y ). The infimum ov er thos e D > 1 for which ( M , d M ) em beds with distortion D into ( Y , d Y ) is denoted by c Y ( M ). If ( M , d M ) do es no t admit a bi- Lipschitz embedding into ( Y , d Y ), we will write c Y ( M ) = ∞ . Throughout this pap er, for p > 1, the space L p will stand for L p ([0 , 1] , λ ), where λ is Lebe sgue measure. The spaces ℓ p and ℓ n p will stand for the space of p -summable infinite s e q uences, and R n equipp e d with the ℓ p norm, resp ectively . Much o f this pap er will deal with bi-Lipschitz embedding s of fin ite metric spaces int o L p . Since every n -p oint subset of an L p (Ω , µ ) s pa ce embeds isometrica lly into ℓ n ( n − 1) / 2 p (see the discussion in [12]), when it comes to em bedding s of finite metric spaces, the distinctio n b etw een different L p (Ω , µ ) spaces is ir relev an t. Nevertheless, later, in the study of the embeddability of the Heisenberg group, we will need to distinguish be tw een sequence spac es and function spaces. F or p > 1 w e will use the shorter nota tion c p ( M ) = c L p ( M ). The par ameter c 2 ( M ) is known as the E uclidean distortio n of M . Dv oretzky’s theorem says tha t if Y is a n infinite dimensional B a nach space then c Y ( ℓ n 2 ) = 1 for all n ∈ N . Th us, for every finite metric spa ce M and every infinite dimensional B anach space Y , we hav e c 2 ( M ) > c Y ( M ). The following famous theorem of Bourgain [15] will play a key r ole in what follows: Theorem 2.1 (Bourgain’s embedding theo rem [15]). F or every n - p oint metric sp ac e ( M , d M ) , we have c 2 ( M ) . log n. (1) Bourga in proved in [15] that the estimate (1) is sharp up to an iter ated loga - rithm factor, i.e., that there exist arbitra rily larg e n - p o int metric spaces M n for which c 2 ( M n ) & log n log log n . The log log n term was remov ed in the imp ortant pa- per [5 6] of Linial, London and Rabino vic h, who sho w ed that the shor tes t pa th met- ric on b ounded degree n - vertex expander gr aphs has Euclidea n distortio n & lo g n . If one is interested o nly in embeddings into infinite dimensional Banach s paces, then Theorem 2.1 is sta ted in the stro ngest p ossible form: as no ted ab ov e, it implies that for every infinite dimensional Banach s pa ce Y , we hav e c Y ( M ) . log n . Below, we will actually us e Theorem 2 .1 fo r e mbedding s into L 1 , i.e., we will use the fact that c 1 ( M ) . log n . The expa nder based low er bo und o f Linial, London and Rabinovich [56] extends to embeddings into L 1 as well, i.e., even this weaker form o f Bourg ain’s embedding theorem is asymptotica lly sharp. W e re fer to [58, Ch. 1 5] for a comprehensive discussion of these issues, as well as a nice presen tation of the pro of of Bour gain’s embedding theo rem. 4 Assaf Naor 3. L 1 as a metric space Let (Ω , µ ) be a measure space. Define a mapping T : L 1 (Ω , µ ) → L ∞ (Ω × R , µ × λ ), where λ is Leb esgue mea s ure, by: T ( f )( ω , x ) def =    1 0 < x 6 f ( ω ) , − 1 f ( ω ) < x < 0 , 0 otherwise . F or all f , g ∈ L 1 (Ω , µ ) we hav e:    T ( f )( ω , x ) − T ( g )( ω , x )    =  1 g ( ω ) < x 6 f ( ω ) or f ( ω ) < x 6 g ( ω ) , 0 otherwis e . Thu s, for all p > 0 we hav e, k T ( f ) − T ( g ) k p L p (Ω × R ,µ × λ ) = Z Ω Z ( g ( ω ) , f ( ω )] ⊔ ( f ( ω ) ,g ( ω )] dλ ! dµ ( ω ) = Z Ω | f ( ω ) − g ( ω ) | dµ ( ω ) = k f − g k L 1 (Ω ,µ ) . (2) Spec ia lizing (2) to p = 2, we see that: k T ( f ) − T ( g ) k L 2 (Ω × R ,µ × λ ) = q k f − g k L 1 (Ω ,µ ) . Corollary 3. 1. The metric sp ac e  L 1 (Ω , µ ) , k f − g k 1 / 2 L 1 (Ω ,µ )  admits an isometric emb e dding into Hilb ert sp ac e. Another us eful corollar y is obtained when (2) is sp ecialized to the case p = 1. T ake an arbitra ry finite subset X ⊆ L 1 (Ω , µ ). F or every ( ω , x ) ∈ Ω × R consider the set S ( ω , x ) = { f ∈ X : x 6 f ( ω ) } ⊆ X . F or every S ⊆ X w e ca n define a measurable subset E S = { ( ω , x ) ∈ Ω × R : S ( ω , x ) = S } ⊆ Ω × R . By the definition of T , for every f , g ∈ X w e hav e k f − g k L 1 (Ω ,µ ) (2) = k T ( f ) − T ( g ) k L 1 (Ω × R ,µ × λ ) = Z Ω × R    1 S ( w ,x ) ( f ) − 1 S ( w ,x ) ( g )    d ( µ × λ )( ω , x ) = X S ⊆ X ( µ × λ )( E S )    1 S ( f ) − 1 S ( g )    , where here, and in what follows, 1 S ( · ) is the characteristic function o f S . W riting β S = ( µ × λ )( E S ), we hav e the following imp or tant corollar y: Corollary 3.2. L et X ⊆ L 1 (Ω , µ ) b e a finite subset of L 1 (Ω , µ ) . Then ther e exist nonne gativ e numb ers { β S } S ⊆ X ⊆ [0 , ∞ ) such that for al l f , g ∈ X we have: k f − g k L 1 (Ω ,µ ) = X S ⊆ X β S    1 S ( f ) − 1 S ( g )    . (3) Em b eddings of the H eisen b erg group 5 A metric space ( M , d M ) is said to be of ne gative typ e if the metric space  M , d 1 / 2 M  admits an isometric em bedding int o Hilber t space. Such metr ics will play a crucial role in the ensuing discus s ion. This terminolo gy (see e.g., [33]) is due to a classical theorem of Schoenberg [65], which a sserts that ( M , d M ) is of neg ative t ype if and only if for every n ∈ N and every x 1 , . . . , x n ∈ X , the matrix ( d M ( x i , x j )) n i,j =1 is nega tive semidefinite on the orthogo na l complement o f the main diago nal in C n , i.e., for a ll ζ 1 , . . . , ζ n ∈ C with P n j =1 ζ j = 0 we hav e P n i =1 P n j =1 ζ i ζ j d M ( x i , x j ) 6 0. Cor o llary (3.1) can be restated as saying that L 1 (Ω , µ ) is a metric s pace o f negative type. Corollar y (3.2) is often called the cut c one r epr esentation of L 1 metrics . T o explain this terminology , consider the set C ⊆ R n 2 of a ll n × n real matrices A = ( a ij ) such that there is a measur e space (Ω , µ ) a nd f 1 , . . . , f n ∈ L 1 (Ω , µ ) with a ij = k f i − f j k L 1 (Ω ,µ ) for all i, j ∈ { 1 , . . . , n } . If f 1 , . . . , f n ∈ L 1 (Ω 1 , µ 1 ) and g 1 , . . . , g n ∈ L 1 (Ω 2 , µ 2 ) then for all c 1 , c 2 > 0 and i, j ∈ { 1 , . . . , n } we have c 1 k f i − f j k L 1 (Ω 1 ,µ 1 ) + c 2 k f i − f j k L 1 (Ω 2 ,µ 2 ) = k h i − h j k L 1 (Ω 1 ⊔ Ω 2 ,µ 1 ⊔ µ 2 ) , where h 1 , . . . , h n are functions defined on the disjoin t union Ω 1 ⊔ Ω 2 as follows: h i ( ω ) = c 1 f i ( ω ) 1 Ω 1 ( ω ) + c 2 g i ( ω ) 1 Ω 2 ( ω ). This observ ation s hows that C is a cone (of dimension n ( n − 1) / 2). Identit y (3) says that the cone C is genera ted by the rays induced by cut semimetrics, i.e., by ma trices of the form a ij = | 1 S ( i ) − 1 S ( j ) | for some S ⊆ { 1 , . . . , n } . It is not difficult to see that thes e rays are actually the extreme r ays of the cone C . Cara th ´ eodory ’s theor em (for cones) says that we can choose the coefficie n ts { β S } S ⊆ X in (3) so that only n ( n − 1) / 2 of them are non- z ero. 4. The Sparsest Cut Problem Given n ∈ N and tw o symmetric functions C, D : { 1 , . . . , n } × { 1 , . . . , n } → [0 , ∞ ) (called capac ities and demands, resp ectively), and a subset ∅ 6 = S ( { 1 , . . . , n } , write Φ( S ) def = P n i =1 P n j =1 C ( i, j ) · | 1 S ( i ) − 1 S ( j ) | P n i =1 P n j =1 D ( i, j ) · | 1 S ( i ) − 1 S ( j ) | . (4) The v alue Φ ∗ ( C, D ) def = min ∅6 = S ( { 1 ,...,n } Φ( S ) (5) is the minim um ov er a ll c uts (tw o-part partitions) o f { 1 , . . . , n } of the ratio b etw een the total capacity cro ssing the b oundary o f the cut and the total dema nd cros s ing the b oundary of the cut. Finding in p olynomia l time a cut for which Φ ∗ ( C, D ) is attained up to a definite m ultiplicative consta n t is called the Spars est Cut pro ble m. This problem is used as a subr outine in many approximation algo rithms for NP- ha rd pro blems; see the survey articles [68, 22], a s well as [53, 1] a nd the reference s in [6, 5 ] fo r so me of the v ast literature on this topic. Computing Φ ∗ ( C, D ) exa ctly has b een long known to 6 Assaf Naor be NP- hard [6 7]. Mor e recently , it w as s hown in [31] that there exists ε 0 > 0 s uch that it is NP-har d to approximate Φ ∗ ( C, D ) to within a facto r sma ller than 1 + ε 0 . In [4 7, 2 4] it was shown that it is Unique Games har d to approximate Φ ∗ ( C, D ) to within any constan t factor (see [44, 45] for mor e informa tion on the Unique Ga mes Conjecture; we will return to this issue in Section 4.3.3). It is customar y in the literature to hig hlight the supp ort of the ca pacities func- tion C : this allows us to in tro duce a particular y impo r tant sp ecial cas e of the Sparsest Cut Problem. Thus, a differe nt wa y to formulate the ab ove setup is via an n -vertex graph G = ( V , E ), with a po sitive weight (called a capacity) C ( e ) as- so ciated to each edge e ∈ E , and a nonnega tive weigh t (called a demand) D ( u , v ) asso ciated to each pa ir o f vertices u , v ∈ V . The go a l is to ev a luate in p olynomia l time (and in particular , while exa mining only a neg ligible fraction of the s ubs ets of V ) the quantit y : Φ ∗ ( C, D ) = min ∅6 = S ( V P uv ∈ E C ( uv ) | 1 S ( u ) − 1 S ( v ) | P u,v ∈ V D ( u, v ) | 1 S ( u ) − 1 S ( v ) | . T o g et a feeling for the meaning o f Φ ∗ , consider the case C ( e ) = D ( u, v ) = 1 for all e ∈ E and u , v ∈ V . This is an imp ortant instance of the Sparse st Cut problem which is called “Spar sest Cut with Unifor m Demands”. In this case Φ ∗ bec omes: Φ ∗ = min ∅6 = S ( V # { edges joining S and V \ S } | S | · | V \ S | . Thu s, in the case of unifor m demands, the Spars e s t Cut pro blem essentially amounts to solving efficiently the co m binatorial iso p erimetric problem on G : determining the subs e t of the gra ph whose r a tio of edge b oundar y to its size is as small as po ssible. In the literature it is also customa ry to emphasize the size of the supp or t of the demand function D , i.e., to state b ounds in terms of the num ber k of pa irs { i, j } ⊆ { 1 , . . . , n } for which D ( i, j ) > 0 . F or the s ake o f simplicity of exp osition, we will not adopt this c o nv en tion here, and state all of our b ounds in terms o f n r ather than the num ber of pos itive dema nd pair s k . W e refer to the relev ant references for the simple mo difica tio ns that are r equired to obta in b o unds in terms of k alone. ¿F rom now on, the Spa r sest Cut problem will b e understo o d to b e with genera l capacities a nd demands; when disc us sing the sp ecial case of uniform dema nds we will say so explicitly . In applications, gener al capa cities and dema nds are used to tune the notion o f “interface” b etw een S a nd V \ S to a wide v ar iety of co m binatorial optimization pr oblems, which is one of the r easons why the Sparsest Cut pro blem is so versatile in the field o f approximation algor ithms. 4.1. Reform ulation as an optimization problem o v er L 1 . Al- though the Spar sest Cut Pro blem clearly has geometric flav or as a discrete iso p e ri- metric problem, the following key refor mulation o f it, due to [1 1 , 56], ex plicitly relates it to the geo metry o f L 1 . Em b eddings of the H eisen b erg group 7 Lemma 4. 1. Given symmet r ic C , D : { 1 , . . . , n } × { 1 , . . . , n } → [0 , ∞ ) , we have: Φ ∗ ( C, D ) = min f 1 ,...,f n ∈ L 1 P n i =1 P n j =1 C ( i, j ) k f i − f j k 1 P n i =1 P n j =1 D ( i, j ) k f i − f j k 1 . (6) Pr o of. Let φ denote the rig ht hand side of (6), and write Φ ∗ = Φ ∗ ( C, D ). Given a subset S ⊆ { 1 , . . . , n } , by co nsidering f i = 1 S ( i ) ∈ { 0 , 1 } ⊆ L 1 we see that that φ 6 Φ ∗ . In the reverse directio n, if X = { f 1 , . . . , f n } ⊆ L 1 then let { β S } S ⊆ X be the non-neg ative weight s from Coro llary 3.2. F o r S ⊆ X define a subs e t of { 1 , . . . , n } by S ′ = { i ∈ { 1 , . . . , n } : f i ∈ S } . It follows from the definitio n of Φ ∗ that for all S ⊆ X we hav e, n X i =1 n X j =1 C ( i, j ) | 1 S ( f i ) − 1 S ( f j ) | (4) = Φ( S ′ ) n X i =1 n X j =1 D ( i, j ) | 1 S ( f i ) − 1 S ( f j ) | (5) > Φ ∗ n X i =1 n X j =1 D ( i, j ) | 1 S ( f i ) − 1 S ( f j ) | . (7) Thu s n X i =1 n X j =1 C ( i, j ) k f i − f j k 1 (3) = X S ⊆ X β S n X i =1 n X j =1 C ( i, j ) | 1 S ( f i ) − 1 S ( f j ) | (7) > Φ ∗ X S ⊆ X β S n X i =1 n X j =1 D ( i, j ) | 1 S ( f i ) − 1 S ( f j ) | (3) = n X i =1 n X j =1 D ( i, j ) k f i − f j k 1 . It follows that φ > Φ ∗ , as requir ed. 4.2. The linear program. Lemma 4.1 is a reformulation of the Sparsest Cut Problems in terms of a c o ntin uo us optimization pro blem on the space L 1 . Being a reformulation, it sho ws in pa rticular that solving L 1 optimization pr oblems such as the right hand side of (6) is NP -hard. In the beautiful pa pe r [53] of Leigh ton a nd Rao it w as s hown that there exists a po lynomial time a lgorithm that, given an n - vertex gra ph G = ( V , E ), c omputes a nu mber which is guaranteed to be within a factor of . log n o f the uniform Spa rsest Cut v alue (4) . The Leig hton-Rao algorithm uses co mbinatorial ideas which do not apply to Sparse s t Cut with general demands. A breakthrough result, due to Linial- London-Rabinovich [56] a nd Aumann-Rabani [9 ], intro duced embedding metho ds to this field, yielding a p olyno mial time algo rithm which computes Φ ∗ ( C, D ) up to a factor . log n fo r all C, D : { 1 , . . . , n } × { 1 , . . . , n } → [0 , ∞ ). The key idea of [56, 9 ] is based on replacing the finite subset { f 1 , . . . , f n } of L 1 in (6) by an arbitr ary semimetric on { 1 , . . . , n } . Spec ific a lly , by homo- geneity we can always as s ume that the deno minator in (6) equals 1 , in which case Lemma 4.1 s ays that Φ ∗ ( C, D ) equals the minim um of P n i =1 P n j =1 C ( i, j ) d ij , given that P n i =1 P n j =1 D ( i, j ) d ij = 1 and there exist f 1 , . . . , f n ∈ L 1 for which 8 Assaf Naor d ij = k f i − f j k 1 for all i, j ∈ { 1 , . . . , n } . W e can now ignore the fact that d ij was a semimetric that ca me fr om a subset o f L 1 , i.e., we can define M ∗ ( C, D ) to b e the minim um of P n i =1 P n j =1 C ( i, j ) d ij , given that P n i =1 P n j =1 D ( i, j ) d ij = 1, d ii = 0, d ij > 0, d ij = d j i for all i, j ∈ { 1 , . . . , n } ( n ( n − 1) / 2 symmetry constra ints) a nd d ij 6 d ik + d kj for all i, j, k ∈ { 1 , . . . , n } ( 6 n 3 triangle inequality co nstraints). Clearly M ∗ ( C, D ) 6 Φ ∗ ( C, D ), since we are minimizing ov er all semimetr ic s rather than just those a r ising from subsets of L 1 . Mo reov er, M ∗ ( C, D ) can b e computed in p olynomial time up to arbitr a rily go o d precision [40], since it is a linear progr am (minimizing a linear functional in the v ar iables ( d ij ) sub ject to po lynomially many linear co nstraints). The linear prog ram pro duces a se mimetr ic d ∗ ij on { 1 , . . . , n } which satisfies M ∗ ( C, D ) = P n i =1 P n j =1 C ( i, j ) d ∗ ij and P n i =1 P n j =1 D ( i, j ) d ∗ ij = 1 (igno ring arbi- trarily small err ors). By Lemma 4.1 we nee d to somehow rela te this semimetric to L 1 . It is a t this juncture that we see the p ower of Bo urgain’s embedding theo- rem 2.1: the constr aints of the linear progra m only provide us the infor mation that d ∗ ij is a semimetric, a nd nothing else. So, we need to b e able to so mehow handle arbitrar y metr ic spaces—pr ecisely what Bourga in’s theor em do es, by furnishing f 1 , . . . , f n ∈ L 1 such that for all i , j ∈ { 1 , . . . , n } we hav e d ∗ ij log n . k f i − f j k 1 6 d ∗ ij . (8) Now, Φ ∗ ( C, D ) (6) 6 P n i =1 P n j =1 C ( i, j ) k f i − f j k 1 P n i =1 P n j =1 D ( i, j ) k f i − f j k 1 (8) . log n · P n i =1 P n j =1 C ( i, j ) d ∗ ij P n i =1 P n j =1 D ( i, j ) d ∗ ij = log n · M ∗ ( C, D ) . (9) Thu s, Φ ∗ ( C,D ) log n . M ∗ ( C, D ) 6 Φ ∗ ( C, D ), i.e., the p olynomial time algorithm of computing M ∗ ( C, D ) is gura n teed to pr o duce a num ber which is w ithin a factor . log n of Φ ∗ ( C, D ). Remark 4.2. In the a b ove arg umen t we only dis c ussed the alg orithmic task o f fast estimation of the num b er Φ ∗ ( C, D ), r ather than the pr oblem of pr o ducing in p olynomia l time a subset ∅ 6 = S ( { 1 , . . . , n } for whic h Φ ∗ ( S ) is close up to a certain m ultiplicative guar antee to the optimum v alue Φ ∗ ( C, D ). All the algorithms discussed in this pap er pro duce such a s et S , rather than just approximating the nu mber Φ ∗ ( C, D ). In o rder to mo dify the ar gument ab ov e to this setting, one needs to go into the pro of o f Bourga in’s embedding theorem, which as c ur rently stated as just an existential r esult fo r f 1 , . . . , f n as in (8). T his iss ue is addressed in [5 6], which provides a n alg orithmic version of Bour gain’s theorem. Ensuing algorithms in this pap er can b e similarly mo dified to pro duce a go o d cut S , but we will ignore this issue from now on, and co nt in ue to fo cus solely on alg orithms for a pproximate computation of Φ ∗ ( C, D ). Em b eddings of the H eisen b erg group 9 4.3. The semidefi nite program. W e have alrea dy stated in Section 2 that the log arithmic loss in the applica tion (8) of Bourg ain’s theore m ca nno t b e improv ed. Thus, in o rder to obtain a p oly nomial time algorithm with approxi- mation gua rantee b etter than . log n , we need to imp ose additiona l g eometric restrictions on the metric d ∗ ij ; conditions that will hop efully yield a class o f met- ric spaces for which one can prove an L 1 distortion b ound that is a symptotically smaller than the . log n of Bourg ain’s embedding theor em. This is indeed p ossi- ble, based on a quadratic v a riant of the discussio n in Section 4.2; a n approach due to Go emans and Linial [37, 55, 5 4]. The idea o f Go emans and Linia l is based on Co r ollary 3.1, i.e ., o n the fact that the metric space L 1 is of negative type. W e define M ∗∗ ( C, D ) to b e the minimum of P n i =1 P n j =1 C ( i, j ) d ij , sub ject to the constraint that P n i =1 P n j =1 D ( i, j ) d ij = 1 and d ij is a s e mimetr ic of negative type o n { 1 , . . . , n } . The latter c o ndition can b e equiv ale ntly r estated as the req uirement that, in addition to d ij being a semimetric on { 1 , . . . , n } , there exist vectors v 1 , . . . , v n ∈ L 2 such that d ij = k v i − v j k 2 2 for all i, j ∈ { 1 , . . . , n } . Equiv alen tly , ther e exists a symmetric p o sitive semidefinite n × n matrix ( a ij ) (the Gram matrix of v 1 , . . . , v n ), such that d ij = a ii + a j j − 2 a ij for all i, j ∈ { 1 , . . . , n } . Thu s, M ∗∗ ( C, D ) is the minimum o f P n i =1 P n j =1 C ( i, j )( a ii + a j j − 2 a ij ), a linear function in the v ariables ( a ij ), sub ject to the constraint that ( a ij ) is a symmetric p ositive s emidefinite matrix, in conjunction with the linear co nstraints P n i =1 P n j =1 D ( i, j )( a ii + a j j − 2 a ij ) = 1 and for all i , j, k ∈ { 1 , . . . , n } , the tr iangle inequality constraint a ii + a j j − 2 a ij 6 ( a ii + a kk − 2 a ik ) + ( a kk + a j j − 2 a kj ). Such an optimization problem is ca lled a semidefinite pro gram, a nd by the metho ds describ ed in [40], M ∗∗ ( C, D ) can be co mputed with arbitrar ily g o o d prec ision in po lynomial time. Corollar y 3.1 and Le mma 4.1 imply that M ∗ ( C, D ) 6 M ∗∗ ( C, D ) 6 Φ ∗ ( C, D ). The following breakthr o ugh result of Arora, Rao and V azira ni [6] shows that for Sparsest Cut with uniform demands the Go emans-Linial appr oach do es indeed yield an improv ed approximation algorithm: Theorem 4.3 ([6]). In t he c ase of uniform demands, i.e., if C ( i , j ) ∈ { 0 , 1 } and D ( i, j ) = 1 for al l i, j ∈ { 1 , . . . , n } , we have Φ ∗ ( C, D ) √ log n . M ∗∗ ( C, D ) 6 Φ ∗ ( C, D ) . (10) In the cas e of g eneral demands we hav e a lmost the same result, up to lower order factors: Theorem 4.4 ([5]). F or al l symmetric C , D : { 1 , . . . , n } × { 1 , . . . , n } → [0 , ∞ ) we have Φ ∗ ( C, D ) (log n ) 1 2 + o (1) . M ∗∗ ( C, D ) 6 Φ ∗ ( C, D ) . (11) The o (1) term in (11) is . log log log n log log n . W e conjecture that it could be remov ed altogether, though a t pre s ent it se e ms to b e an inher ent a rtifact o f complica tions in the pro of in [5]. 10 Assaf Naor Before explaining s ome o f the ideas b ehind the pr o ofs o f Theorem 4.3 and Theorem 4.4 (the full details are quite le ngthy and are beyond the scop e of this survey), we pro ve, fo llowing [58, Pr op. 15.5.2], a crucia l ident it y (attributed in [58] to Y. Rabinovich) which refo rmulates these r esults in terms of an L 1 embeddability problem. Lemma 4. 5. We have sup  Φ ∗ ( C, D ) M ∗∗ ( C, D ) : C, D : { 1 , . . . , n } × { 1 , . . . , n } → (0 , ∞ )  = sup n c 1  { 1 , . . . , n } , d  : d is a metric of negative type o . (12) Pr o of. The pro of of the fact tha t the left hand side of (12) is at most the r ight hand side of (12) is identical to the way (9) was deduced fro m (8). In the reverse direction, let d ∗ be a metric of negative t yp e o n { 1 , . . . , n } for which c 1 ( { 1 , . . . , n } , d ∗ ) def = c is ma x imal a mong all such metrics. Let C ⊆ R n 2 be the cone in the spa c e of n × n symmetric matrice s from the la st para graph of Sectio n 3, i.e., C consists of all matrices of the form ( k f i − f j k 1 ) for so me f 1 , . . . , f n ∈ L 1 . Fix ε ∈ (0 , c − 1 ) and let K ε ⊆ R n 2 be the set of all symmetric matr ices ( a ij ) for which there exists s > 0 such that sd ∗ ( i, j ) 6 a ij 6 ( c − ε ) sd ∗ ( i, j ) for all i, j ∈ { 1 , . . . , n } . By the definition o f c , the co nv ex sets C and K ε are disjo int, since o therwise d ∗ would admit an embedding into L 1 with distortion c − ε . It follows that there exists a symmetric matrix ( h ε ij ) ∈ R n 2 \ { 0 } and α ∈ R , s uch that P n i =1 P n j =1 h ε ij a ij 6 α for a ll ( a ij ) ∈ K ε , and P n i =1 P n j =1 h ε ij b ij > α for a ll ( b ij ) ∈ C . Since both C a nd K ε are c losed under mult iplication b y p ositive sca lars, necessarily α = 0. Define C ε ( i, j ) def = h ε ij 1 { h ε ij > 0 } and D ε ( i, j ) def = | h ε ij | 1 { h ε ij 6 0 } . By definition of M ∗∗ ( C ε , D ε ), n X i =1 n X j =1 C ε ( i, j ) d ∗ ij > M ∗∗ ( C ε , D ε ) · n X i =1 n X j =1 D ε ( i, j ) d ∗ ij . (13) By cons ide r ing a ij def =  ( c − ε ) 1 { h ε ij > 0 } + 1 { h ε ij < 0 }  d ∗ ( i, j ) ∈ K ε , the inequality P n i =1 P n j =1 h ε ij a ij 6 0 b ecomes: n X i =1 n X j =1 D ε ( i, j ) d ∗ ij > ( c − ε ) n X i =1 n X j =1 C ε ( i, j ) d ∗ ij . (14) A co mbin ation o f (13) and (14) implies that ( c − ε ) M ∗∗ ( C ε , D ε ) 6 1. At the same time, for a ll f 1 , . . . , f n ∈ L 1 , the inequality P n i =1 P n j =1 h ε ij k f i − f j k 1 > 0 is the same as P n i =1 P n j =1 C ε ( i, j ) k f i − f j k 1 > P n i =1 P n j =1 D ε ( i, j ) k f i − f j k 1 , which by Lemma 6 means that Φ ∗ ( C ε , D ε ) > 1 . Thus Φ ∗ ( C ε , D ε ) / M ∗∗ ( C ε , D ε ) > c − ε , and since this holds for a ll ε ∈ (0 , c − 1), the pro of of Lemma 4.5 is complete. Em b eddings of the H eisen b erg group 11 In the c a se of Spars est Cut with uniform demands, we ha ve the follo wing result which is analogo us to Lemma 4.5, where the L 1 bi-Lipschitz disto rtion is replaced by the smalles t p ossible factor by which 1 -Lipschitz functions into L 1 can dis tort the aver age distanc e . The pro of is a slight v aria n t of the pro of of Lemma 4.5; the simple details are left to the reader. This connection betw een Spar sest Cut with uniform dema nds and em bedding s that preser ve the av erage distance is due to Rabinovic h [63]. Lemma 4.6. The supr emum of Φ ∗ ( C, D ) / M ∗∗ ( C, D ) over al l inst anc es of uniform demands, i.e., when C ( i, j ) ∈ { 0 , 1 } and D ( i, j ) = 1 for al l i , j ∈ { 1 , . . . , n } , e quals the infimu m over A > 0 such that for al l metrics d on { 1 , . . . , n } of n e gative typ e, ther e exist f 1 , . . . , f n ∈ L 1 satisfying k f i − f j k 1 6 d ( i, j ) for al l i, j ∈ { 1 , . . . , n } and A P n i =1 P n j =1 k f i − f j k 1 > P n i =1 P n j =1 d ( i, j ) . 4.3.1. L 2 em b eddings of negativ e t yp e metrics. The pr o of of Theo r em 4.3 in [6] is based on a cle ver geometric pa rtitioning pro cedure for metrics of nega- tive type. B uilding heavily on ideas of [6], in conjunction with some substantial additional co m binatorial ar guments, an a lternative a pproach to Theore m 4.3 was obtained in [59], based on a purely graph theoretical statement which is of indep en- dent interest. W e sha ll now sketc h this approach, since it is mo dula r and general, and as suc h it is useful for additional g e ometric corolla ries. W e refer to [59] for more information on these additio na l applications, as well as to [6] for the orig inal pro of of Theorem 4.3. Let G = ( V , E ) b e an n -vertex gra ph. The vertex exp ansion of G , denoted h ( G ), is the largest h > 0 such that every S ⊆ V with | S | 6 n/ 2 has at least h | S | neighbors in V \ S . The e dge exp ansion of G , denoted α ( G ), is the larges t α > 0 such that for every S ⊆ V with | S | 6 n/ 2, the num ber of edg e s joining S and V \ S is at least α | S | · | E | n . The main combinatorial statement of [5 9] relates these tw o notions of expansion of gra phs: Theorem 4.7 (Edge Replacement Theo rem [59]). F or every gr ap h G = ( V , E ) with h ( G ) > 1 2 ther e is a set of e dges E ′ on V with α ( V , E ′ ) & 1 , and such t hat for every uv ∈ E ′ we have d G ( u, v ) . p log | V | . H er e d G is t he shortest p ath metric on G (with r esp e ct to the original e dg e set E ), and al l implicit c onstants ar e universal. It is shown in [59] that the . √ log n b ound on the length of the new edge s in Theorem 4.7 is asympto tica lly tight. The pro of of Theorem 4.7 is inv olved, a nd cannot b e describ ed here: it has tw o comp onents, a combinatorial construction, as well a purely Hilb ertian geometric arg ument based on, and simpler than, the orig - inal alg orithm of [6]. W e shall now explain how Theo rem 4.7 implies Theorem 4 .3 (this is somewhat different from the deduction in [5 9], which deals with a different semidefinite progr am for Sparse s t Cut with uniform demands). Pr o of of The or em 4.3 assuming The or em 4.7. An application of (the ea sy direc- tion of ) Lemma 4 .6 s hows that in or der to prove Theorem 4.3 it suffices to show that if ( M , d ) is an n -p oint metric space of nega tive type, with 1 n 2 P x,y ∈ M d ( x, y ) = 1 , then there exists a mapping F : M → R which is 1-Lipschitz and such that 12 Assaf Naor 1 n 2 P x,y ∈ M | F ( x ) − F ( y ) | & 1 / √ log n . In w ha t follows we use the s tandard nota - tion for closed balls : for x ∈ M and t > 0, set B ( x, t ) = { y ∈ M : d ( x, y ) 6 t } . Cho ose x 0 ∈ M with 1 n P y ∈ M d ( x 0 , y ) = r def = min x ∈ M 1 n P y ∈ M d ( x, y ). Then r 6 1 n 2 P x,y ∈ M d ( x, y ) = 1, implying 1 > 1 n P y ∈ M d ( x 0 , y ) > 2 n | M \ B ( x 0 , 2) | , o r | B ( x 0 , 2) | > n/ 2 . Similarly | B ( x 0 , 4) | > 3 n/ 4. Assume first that 1 n 2 P x,y ∈ B ( x 0 , 4) d ( x, y ) 6 1 4 (this will b e the easy cas e ). Then 1 = 1 n 2 X x,y ∈ M d ( x, y ) 6 1 4 + 2 n 2 X x ∈ M X y ∈ M \ B ( x 0 , 4)  d ( x, x 0 ) + d ( x 0 , y )  = 1 4 + 2 r n | M \ B ( x 0 , 4) | + 2 n X y ∈ M \ B ( x 0 , 4) d ( x 0 , y ) 6 3 4 + 2 n X y ∈ M \ B ( x 0 , 4) d ( x 0 , y ) , or 1 n P y ∈ M \ B ( x 0 , 4) d ( x 0 , y ) > 1 8 . Define a 1- L ipschitz mapping F : M → R by F ( x ) = d  x, B ( x 0 , 2)  = min y ∈ B ( x 0 , 2) d ( x, y ). The tr iangle inequa lit y implies that for every y ∈ M \ B ( x 0 , 4) we hav e F ( y ) > 1 2 d ( y , x 0 ). Thus 1 n 2 X x,y ∈ M | F ( x ) − F ( y ) | > | B ( x 0 , 2) | n 2 X y ∈ M \ B ( x 0 , 4) d  y , B ( x 0 , 2)  > 1 2 n X y ∈ M \ B ( x 0 , 4) 1 2 d ( y , x 0 ) & 1 = 1 n 2 X x,y ∈ M d ( x, y ) . This completes the ea sy ca se, where there is even no loss of 1 / √ log n (and we did not use yet the ass umption that d is a metric of nega tive type). W e ma y ther efore assume from now o n that 1 n 2 P x,y ∈ B ( x 0 , 4) d ( x, y ) > 1 4 . The fact that d is of negative t yp e means that there ar e vectors { v x } x ∈ M ⊆ L 2 such that d ( x, y ) = k v x − v y k 2 2 for all x, y ∈ M . W e will show that for a small eno ugh universal co nstant ε > 0, there are tw o sets S 1 , S 2 ⊆ B ( x 0 , 4) such that | S 1 | , | S 2 | > εn and d ( S 1 , S 2 ) > ε 2 / √ log n . O nce this is achieved, the ma pping F : M → R given by F ( x ) = d ( x, S 1 ) will satisfy 1 n 2 P x,y ∈ M | F ( x ) − F ( y ) | > 2 n 2 | S 1 | · | S 2 | ε 2 √ log n > 2 ε 4 √ log n , as desired. Assume for contradiction that no such S 1 , S 2 exist. Define a set of edges E 0 on B ( x 0 , 4) by E 0 def = n { x, y } ⊆ B ( x 0 , 4) : x 6 = y ∧ d ( x, y ) < ε 2 / √ log n o . Our cont rap ositive assumption says that a ny tw o subsets S 1 , S 2 ⊆ B ( x 0 , 4) with | S 1 | , | S 2 | > εn > ε | B ( x 0 , 4) | are joined by an edge from E 0 . By a (simple) genera l graph theoretica l lemma (see [59, Lem 2.3 ]), this implies that, provided ε 6 1 / 1 0, there exists a subset V ⊆ B ( x 0 , 4) with | V | > (1 − ε ) | B ( x 0 , 4) | & n , such tha t the graph induced by E 0 on V , i.e., G =  V , E = E 0 ∩  V 2   , has h ( G ) > 1 2 . W e ar e now in p osition to apply the Edge Replacement Theorem, i.e., Theo- rem 4.7. W e obtain a new set of edg es E ′ on V s uch that α ( V , E ′ ) & 1 a nd for every xy ∈ E ′ we hav e d G ( x, y ) . √ log n . The la tter co ndition means that there exists a pa th { x = x 0 , x 1 , . . . , x m = y } ⊆ V such that m . √ log n and x i x i − 1 ∈ E Em b eddings of the H eisen b erg group 13 for every i ∈ { 1 , . . . , m } . By the definition of E , this implies that xy ∈ E ′ = ⇒ d ( x, y ) 6 n X i =1 d ( x i , x i − 1 ) 6 m ε 2 √ log n . ε 2 . (15) It is a standard fact (the equiv alenc e b etw een edge expansion and a Cheeger inequality) that for every f : V → L 1 we hav e 1 | E ′ | X xy ∈ E ′ k f ( x ) − f ( y ) k 1 > α ( V , E ′ ) 2 | V | 2 X x,y ∈ V k f ( x ) − f ( y ) k 1 . (16) F or a pro of o f (16 ) see [59, F a ct 2.1]: this is a simple consequence of the cut cone representation, i.e., Corollar y 3.2, since the identit y (3) shows that it suffices to prov e (16) when f ( x ) = 1 S ( x ) for s o me S ⊆ V , in whic h ca se the desir ed ineq uality follows immediately from the definition o f the edge expansio n α ( V , E ′ ). Since L 2 is isometr ic to a subset of L 1 (see, e.g., [71]), it follows from (16) and the fact that α ( V , E ′ ) & 1 that ε (15) & 1 | E ′ | X xy ∈ E ′ p d ( x, y ) = 1 | E ′ | X xy ∈ E ′ k v x − v y k 2 & 1 | V | 2 X x,y ∈ V k v x − v y k 2 & 1 n 2 X x,y ∈ V p d ( x, y ) . (17 ) Now comes the po in t where we use the assumption 1 n 2 P x,y ∈ B ( x 0 , 4) d ( x, y ) > 1 4 . Since for any x, y ∈ B ( x 0 , 4) we hav e d ( x, y ) 6 8, it follows that the num ber of pa ir s ( x, y ) ∈ B ( x 0 , 4) × B ( x 0 , 4) with d ( x, y ) > 1 / 8 is at least n 2 / 64. Since | V | > (1 − ε ) | B ( x 0 , 4) | , the n um be r of suc h pairs whic h ar e also in V × V is a t least n 2 64 − 3 εn 2 & n 2 , provided ε is small enoug h. Thus 1 n 2 P x,y ∈ V p d ( x, y ) & 1, and (17) bec o mes a contradiction for s ma ll e no ugh ε . Remark 4. 8. The ab ov e pro of of Theorem 4 .7 used v ery little of the fact tha t d is a metric of negative type. In fact, a ll that w as required was that d admits a quasisymmetric embedding into L 2 ; see [59]. It remains to s ay a few words ab out the pro of of Theore m 4 .4. Unfortunately , the present pro of of this theo r em is long a nd inv olv ed, and it relies o n a v arie ty of results from metric embedding theory . It would b e of interest to obtain a simpler pro of. Lemma 4.5 implies that Theore m 4.4 is a consequence of the following embedding result: Theorem 4. 9 ([5]). Every n -p oint metric sp ac e of ne ga tive typ e emb e ds into Hilb ert sp ac e with distortion . (log n ) 1 2 + o (1) . Theorem 4.9 improves ov er the previously known [23] b ound of . (log n ) 3 / 4 on the Euclidean distortion of n -p oint metric spaces of negative type. As w e shall explain b elow, Theorem 4.9 is tight up to the o (1) term. The pro of of Theorem 4.9 us e s the following notion from [5]: 14 Assaf Naor Definition 4 .10 (Random zero-sets [5 ]). Fix ∆ , ζ > 0, a nd p ∈ (0 , 1). A metric space ( M , d ) is s aid to admit a rando m zero set at sca le ∆, whic h is ζ -spreading with probability p , if there is a proba bilit y dis tribution µ ov er subsets Z ⊆ M such that µ ( { Z : y ∈ Z ∧ d ( x, Z ) > ∆ /ζ } ) > p for every x, y ∈ M with d ( x, y ) > ∆. W e denote by ζ ( M ; p ) the least ζ > 0 such that fo r every ∆ > 0, M admits a random zero set at scale ∆ w hich is ζ -s preading with probability p . The connection to metrics of negative type is due to the following theorem, which can b e viewed as the ma in structural consequence of [6]. Its pro of use s [6] in conjunction with tw o additiona l ing redients: an a nalysis of the algorithm of [6] due to [50], and a clever iterative applica tio n of the alg orithm of [6], due to [2 3], while carefully reweighting p oints a t each step. Theorem 4 .11 (Rando m zero sets for negative t yp e metrics ). Ther e exists a universal c onstant p > 0 su ch that any n -p oint metric sp ac e ( M , d ) of ne gative typ e satisfies ζ ( M ; p ) . √ log n . Random zer o s e ts are related to embeddings as follows. Fix ∆ > 0. Let ( M , d ) be a finite metric s pa ce, and fix S ⊆ M . B y the definition of ζ ( S ; p ), ther e exists a distribution µ ov er subsets Z ⊆ S such that for e very x, y ∈ S with d ( x, y ) > ∆ we hav e µ ( { Z ⊆ S : y ∈ Z ∧ d ( x, Z ) > ∆ /ζ ( S ; p ) } ) > p . Define ϕ S, ∆ : M → L 2 ( µ ) by ϕ S, ∆ ( x ) = d ( x, Z ). Then ϕ S, ∆ is 1-Lipschitz, and for every x, y ∈ S with d ( x, y ) > ∆, k ϕ S, ∆ ( x ) − ϕ S, ∆ ( y ) k L 2 ( µ ) =  Z 2 S [ d ( x, Z ) − d ( y , Z )] 2 dµ ( Z )  1 / 2 > ∆ √ p ζ ( S ; p ) . (18) The remaining task is to “ glue” the mapping s { ϕ S, ∆ : ∆ > 0 , S ⊆ M } to for m an embedding of M in to Hilb ert space with the distortion claimed in Theorem 4 .9. A key ingredient of the pr o of of Theorem 4.9 is the embedding metho d ca lled “Mea sured Descent”, that was developed in [48]. The res ults of [48] were sta ted as embedding theor ems rather than a g luing pro cedure; the realiza tion that a part of the a r guments of [48] can b e form ulated explicitly as a gener a l “ gluing lemma” is due to [5 0]. In [5] it was necessary to enhance the Measured Descent techn ique in order to prov e the following key theorem, which together with (18) and Theo r em 4.11 implies Theo rem 4.9. See als o [4] for a different enhancement of Measured Descent, which also implies Theo rem 4.9. The pro o f o f Theore m 4.12 is quite intricate; we refer to [5] for the details. Theorem 4.12 . Le t ( M , d ) b e an n -p oi nt metric sp ac e. S upp ose that ther e is ε ∈ [1 / 2 , 1] such t hat for every ∆ > 0 , and every su bset S ⊆ M , ther e exists a 1- Lipschitz map ϕ S, ∆ : M → L 2 with || ϕ S, ∆ ( x ) − ϕ S, ∆ ( y ) || 2 & ∆ / (log | S | ) ε whenever x, y ∈ S and d ( x, y ) > ∆ . Then c 2 ( M ) . (log n ) ε log log n . The following co rollar y is an obvious co nsequence of Theorem 4.9, due to the fact that L 1 is a metric space of neg ative type . Corollary 4.13. Every X ⊆ L 1 emb e ds into L 2 with distortion . (log | X | ) 1 2 + o (1) . Em b eddings of the H eisen b erg group 15 W e stated Corolla r y 4.13 since it is o f sp ecial imp ortance: in 1969, E nflo [3 4] prov ed that the Hamming cub e, i.e., { 0 , 1 } k equipp e d with the metric induced from ℓ k 1 , has Euclidean distortion √ k . Coro llary 4.13 says that up to low er order factors, the Hamming cub e is among the most non-Euclidean subset of L 1 . Ther e are very few known results o f this type, i.e., (almo st) sharp ev aluations o f the largest Euclidean distortio n of an n -p oint subset of a na tur al metric space. A notable such r esult is Matou ˇ s ek’s theorem [57] that any n - po int subset of the infinite bina ry tr e e has Euclide a n distor tion . √ log log n , and c o nsequently , due to [20], the same ho lds tr ue for n -p oint subsets o f, say , the hype r b olic plane. This is tight due to Bo urgain’s matching low er b ound [16] for the Euclidean distortion of finite depth complete binary tr ees. 4.3.2. The Go emans- Li nial conjecture. Theorem 4 .4 is the bes t known ap- proximation algorithm for the Spar sest Cut P r oblem (and Theore m 4.3 is the b est known algor ithm in the ca se of uniform demands). But, a compariso n of Lemma 4.5 and Theorem 4 .9 reveals a po s sible av en ue for further improv emen t: Theorem 4.9 pro duces an embedding of negative t yp e metrics into L 2 (for which the b ound of Theo rem 4.9 is s harp up to low er order facto rs), while for Lemma 4.5 all we need is an em bedding int o the larger space L 1 . It was c o njectured by Go e ma ns and Linial (see [37, 55, 5 4] and [58, pg. 37 9–380 ]) that any finite metric space of negative type embeds into L 1 with distortion . 1 . If true, this would yield, via the Go ema ns-Linial semidefinite relaxation, a constant factor approximation algorithm for Sparsest Cut. As we shall s e e below, it turns out that the Go emans-L inial conjecture is false, and in fact there exis t [30] arbitra r ily large n -p oint metric spaces M n of neg - ative type for which c 1 ( M n ) > (log n ) c , where c is a universal constant. Due to the dua lit y a rgument in Le mma 4.5, this mea ns tha t the alg orithm of Sec- tion 4.3 is do omed to make a n err or of a t least (log n ) c , i.e., ther e exist capacity and demand functions C n , D n : { 1 , . . . , n } × { 1 , . . . , n } → [0 , ∞ ) for which we hav e M ∗∗ ( C n , D n ) . Φ ∗ ( C n , D n ) / (log n ) c . Such a statement is re fer red to in the literature as the fact that the inte gr ality gap of the Go emans-Linial semidefinite relaxatio n of Sparsest Cut is at least (log n ) c . 4.3.3. Unique Games hardness and the Kho t-Vishnoi i n tegralit y gap. Khot’s Unique Ga mes Conjecture [44] is that for every ε > 0 there exists a prime p = p ( ε ) such that there is no p olynomial time algorithm that, g iven n ∈ N and a sys tem o f m -linea r equations in n - v ar ia bles of the for m x i − x j = c ij mo d p for some c ij ∈ N , de ter mines whether there exists an as signment of an integer v alue to each v ariable x i such that a t least (1 − ε ) m of the equations a re satisfied, or whether no ass ignment of s uch v a lues can sa tisfy more than εm of the e q uations (if neither of these p os sibilities o cc ur , then an a rbitrary output is a llow ed). This formulation of the conjecture is due to [46], where it is shown that it is equiv alen t to the orig inal fo r mulation in [44]. The Unique Games Conjecture is by now a common assumption that has numerous applications in computational complexity; see the survey [45] (in this c ollection) for more informatio n. 16 Assaf Naor In [47, 24] it was shown that the existence of a po lynomial time constant factor approximation a lgorithm for Sparsest Cut would refute the Unique Games Conjec- ture, i.e., one can use a p o ly nomial time constant factor approximation algo rithm for Sparsest Cut to solve in p olynomial time the ab ove a lgorithmic task for linear equations. F or a perio d of time in 2004, this computatio nal hardness result led to a strange situation: either the complexity theoretic Unique Games Co njecture is tr ue , or the purely geometric Goe ma ns-Linial conjecture is true, but not b o th. In a remark able tour de force , K hot and Vishnoi [47] delved into the pro o f o f their hardness result and managed to construct from it a co ncrete family of arbitra rily lar ge n -p o int metric spac e s M n of neg ative type for which c 1 ( M n ) & (log log n ) c , wher e c is a universal constant, thus refuting the Go emans-Linia l conjecture. Subsequently , these Kho t- Vis hno i metric spaces M n were analy z ed in [49], re s ulting in the lower bo und c 1 ( M n ) & log log n . F urther work in [32] yie lde d a & log log n in tegrality g ap for Spar s est Cut with uniform dema nds, i.e., “av erage distor tion” L 1 embeddings (in the sense of Lemma 4.6) o f negative type metrics were r uled out as well. 4.3.4. The Bretagnoll e, Dacunha-Castelle, Krivi ne theorem and i n v ari- an t m e trics on Ab el ian groups. A co m bination of Schoenber g’s classical char- acterization [65] o f metric space s tha t are is ometric to subsets of Hilber t space, and a theorem of Bretagnolle, Dacunha-Castelle and Kriv ine [1 8] (see also [70]), implies that if p ∈ [1 , 2] and ( X , k · k X ) is a separa ble Banach space such that the metric space ( X, k x − y k p/ 2 X ) is isometric to a subse t of Hilb ert spa c e , then X is (linear ly) isometric to a subspace of L p . Specia lizing to p = 1 we see that the Go ema ns-Linial conjecture is true for Banach spaces. With this motiv ation for the Go emans-Linial conjecture in mind, one notices that the Go emans-Linia l conjecture is part o f a natural one par ameter family of conjecture s which attempt to extend the theor em Bretagnolle , Dacunha- Castelle and Krivine to g eneral metric spaces rather tha n Banach spaces: is it true that for p ∈ [1 , 2) any metr ic space ( M , d ) for which ( M , d p/ 2 ) is isometric to a subset of L 2 admits a bi-Lipschitz embedding into L p ? This generalized Go emans-Linial conjecture turns out to b e false for all p ∈ [1 , 2); our example based on the Heisenber g group furnishes counter-examples for all p . It is also known that certain inv aria nt metrics on Ab e lian g roups satisfy the Go emans-Linial conjecture: Theorem 4.1 4 ([1 0]). L et G b e a fin ite Ab elian gr oup, e quipp e d with an invariant metric ρ . Supp ose that 2 6 m ∈ N sa tisfies mx = 0 for al l x ∈ G . Denote D = c 2  G, √ ρ  . Then c 1 ( G, ρ ) . D 4 log m . It is an interesting op en ques tion whether the dep endence o n the exp onent m of the group G in Theorem 4.14 is necessa ry . Can one construct a co un ter-example to the Go emans-Linia l c onjecture which is a n inv aria nt metric on the cyclic gr oup C n of o rder n ? Or, is ther e for every D > 1 a co nstant K ( D ) such that for every inv ar iant metric ρ o n C n for which c 2  G, √ ρ  6 D we hav e c 1 ( G, ρ ) 6 K ( D )? One can vie w the ab ov e discussion as motiv a tion for why one might co nsider the Heisenberg group as a p otential counter-example to the Go e mans-Linial con- jecture. Assuming that we ar e interested in in v aria nt metrics on groups, we wish Em b eddings of the H eisen b erg group 17 to depar t fro m the se tting of Ab elian gr o ups or Bana ch spaces, and if a t the sa me time we would like our example to hav e so me useful analytic pr op erties (such as inv ar iance under r escaling and the av ailability of a gro up no rm), the Heisenber g group suggests itself as a natural candidate. This plan is carr ie d out in Section 5. 5. Em b eddings of the Heisen b erg group The purp ose of this section is to discuss Theorem 1.1 and Theorem 1.2 from the int ro duction. Before do ing s o, we have an imp or tant item o f unfinished business : relating the Heisenber g group to the Sparsest Cut P roblem. W e will do this in Section 5.1, following [52]. In pr eparation, w e need to recall the Carnot-Carath´ eo dory geo metry of the con- tin uous Heisenber g group H , i.e., R 3 equipp e d with the no n-commutativ e pro duct ( a, b, c ) · ( a ′ , b ′ , c ′ ) = ( a + a ′ , b + b ′ , c + c ′ + ab ′ − ba ′ ). Due to la ck o f space, this will hav e to b e a crash course , and we refer to the relev an t intro ductory sections of [2 9] for a more thoro ugh discuss io n. The identit y ele men t o f H is e = (0 , 0 , 0), and the in verse element of ( a, b , c ) ∈ H is ( − a, − b , − c ). The center o f H is the z -axis { 0 } × { 0 } × R . F or g ∈ H the horizontal plane at g is defined as H g = g ( R × R × { 0 } ). An affine line L ⊆ H is calle d a horizontal line if for some g ∈ H it passes thr ough g and is co ntained in the a ffine plane H g . The standard scala r pro duct h· , ·i o n H e naturally induces a sca lar pro duct h· , ·i g on H g by h g x, g y i g = h x, y i . Consequen tly , we ca n define the Carnot-Car ath´ eo dor y metric d H on H by letting d H ( g , h ) b e the infimum of lengths of smo oth cur ves γ : [0 , 1] → H s uch that γ (0) = g , γ (1) = h and for all t ∈ [0 , 1] we hav e γ ′ ( t ) ∈ H γ ( t ) (and, the leng th of γ ′ ( t ) is c o mputed with resp ect to the scalar pr o duct h· , ·i γ ( t ) ). The b al l-b ox principle (see [39]) implies that d H  ( a, b, c ) , ( a ′ , b ′ , c ′ )  is b ounded ab ove and b elow b y a co nstant multiple of | a − a ′ | + | b − b ′ | + p | c − c ′ + ab ′ − ba ′ | . Mor eov er, since the integer gr id H ( Z ) is a discrete co co mpa ct subgro up of H , the word metric d W on H ( Z ) is bi-Lipschitz equiv ale nt to the re s triction of d H to H ( Z ) (see, e.g, [1 9]). F or θ > 0 define the dilation op erator δ θ : H → H by δ θ ( a, b, c ) = ( θ a, θ b, θ 2 c ). Then for a ll g , h ∈ H we hav e d H ( δ θ ( g ) , δ θ ( h )) = θ d H ( g , h ). The Leb esgue measure L 3 on R 3 is a Haar measure of H , and the volume of a d H -ball of radius r is pro po rtional to r 4 . 5.1. Heisen berg metrics with isometric L p sno wflak es. F o r ev- ery ( a, b , c ) ∈ H and p ∈ [1 , 2), define M p ( a, b, c ) = 4 p ( a 2 + b 2 ) 2 + 4 c 2 · cos p 2 arccos a 2 + b 2 p ( a 2 + b 2 ) 2 + 4 c 2 !!! 1 /p . It was shown in [52] that M p is a gr oup norm o n H , i.e., fo r all g , h ∈ H a nd θ > 0 we hav e M p ( g h ) 6 M p ( g ) + M p ( h ), M p ( g − 1 ) = M p ( g ) a nd M p ( δ θ ( g )) = θM p ( g ). Thu s d p ( g , h ) def = M p ( g − 1 h ) is a left-inv ariant metric o n H . The metric d p is bi- Lipschitz equiv a lent to d H with distor tion of o rder 1 / √ 2 − p (see [52]). Mor eov er, 18 Assaf Naor it was shown in [52] that ( H , d p/ 2 p ) admits an isometric embedding into L 2 . Thus, in pa rticular, the metric space ( H , d 1 ), which bi-Lips chit z equiv alen t to ( H , d H ), is of ne g ative type. The fact that ( H , d H ) do es not admit a bi-Lipschitz embedding in to L p for any 1 6 p < ∞ will show that the generalized Go ema ns-Linial co njecture (see Section 4 .3.4) is false. In particula r, ( H , d 1 ), a nd hence b y a standard r escaling argument also ( H ( Z ) , d 1 ), is a co unter-example to the Go emans-Linia l conjecture. Note that it is cr ucial here that we ar e dealing w ith the function space L p rather than the sequence space ℓ p , in order to use a compactness ar gument to deduce from this statement that there exist arbitra rily large n -p oint metric spa ces ( M n , d ) such that ( M n , d p/ 2 ) is iso metric to a subset of L 2 , yet lim n →∞ c p ( M n ) = ∞ . The fact that this statement follows from non-embeddability into L p is a cons equence of a well known ultra power arg ument (see [42]), yet for ℓ p this statement is false (e.g., ℓ 2 do es not admit a bi-Lipschitz em be dding into ℓ p , but all finite subsets of ℓ 2 embed isometrica lly into ℓ p ). Unfortunately , this issue crea tes substantial difficulties in the case of prima ry in terest p = 1. In the reflexive range p > 1, or for a sepa rable dual space such as ℓ 1 (= c ∗ 0 ), the non-embedda bility of H fo llows from a natural extensio n of a cla s sical r esult of Pansu [6 1], as we explain in Section 5.2. This a pproach fails badly when it comes to embeddings into L 1 : for this purp ose a novel metho d of Cheeg er a nd Kleiner [25] is needed, a s descr ibe d in Section 5.3. 5.2. P ansu differentiab ilit y . Let X b e a Ba nach space and f : H → X . F ollowing [61 ], f is sa id to hav e a Pansu deriv ativ e at x ∈ H if for every y ∈ H the limit D x f ( y ) def = lim θ → 0  f ( xδ θ ( y )) − f ( x )  /θ exists, a nd D x f : H → X is a group homomorphism, i.e., for all y 1 , y 2 ∈ H we hav e D x f ( y 1 y − 1 2 ) = D x f ( y 1 ) − D y f ( y 2 ). Pansu prov ed [6 1] that every f : H → R n which is L ips chit z in the metr ic d H is Pansu differentiable a lmost everywhere. It was obse r ved in [52, 27] that this result holds true if the target spa ce R n is replace d by any Banach spa ce with the Radon-Nikod ´ ym prop erty , in particula r X can b e any reflexive Banach space such as L p for p ∈ (1 , ∞ ), or a sepa rable dual Banach spa c e such as ℓ 1 . As noted b y Semmes [66], this implies that H do es not a dmite a bi- Lipschitz embedding into any Banach spa ce X with the Radon-Niko d´ ym pr op erty: a bi- Lipschitz condition for f implies that at a p o int x ∈ H of Pansu differentiabilit y , D x f is also bi- Lipschitz, and in par ticular a group iso morphism. But tha t’s imp oss ible since H is non-commutativ e, unlike the a dditive group of X . 5.3. Cheeger-Kleiner differen tiabilit y . Differentiabilit y theorems fail badly when the tar get spa c e is L 1 , even for functions defined on R ; consider Aron- sza jn’s ex ample [3] o f the “moving indica to r function” t 7→ 1 [0 ,t ] ∈ L 1 . F or L 1 - v alued Lipschitz functions o n H , Cheeg er and K leiner [2 5, 28] de velop ed an al- ternative differen tiation theory , which is s ufficie ntly str o ng to s how that H do es not admit a bi-Lips chit z embedding int o L 1 . Roug hly sp eak ing , a differ ent iation theorem states that in the infinitesimal limit, a Lipschitz mapping converges to a mapping that b elongs to a certa in “structured” s ubcla ss of mappings (e.g ., linea r mappings or gr oup homomo rphisms). The Cheeger-K leiner theory shows that, in Em b eddings of the H eisen b erg group 19 a sense that will b e made precise b elow, L 1 -v alued Lips chit z functions on H a re in the infinitesimal limit similar to Ar o nsza jn’s moving indicator . F or an op en subset U ⊆ H let Cut( U ) denote the space of (equiv alence s classes up to measure zero) of measura ble subsets o f U . Let f : U → L 1 be a Lipschitz function. An infinitary v ariant of the cut-cone decomp ositio n of Coro llary 3.2 (see [25]) asser ts that there exists a measure Σ f on Cut( U ), such that for a ll x, y ∈ U we have k f ( x ) − f ( y ) k 1 = R Cut( U ) | 1 E ( x ) − 1 E ( y ) | d Σ f ( E ). The mea sure Σ f is calle d the cut me asure of f . The idea of Cheeger a nd Kleiner is to detect the “infinitesimal regula rity” of f in terms of the infinitesimal b ehavior of the measure Σ f ; mo re pre c isely , in terms of the shap e of the sets E in the s upp or t o f Σ f , a fter passing to an infinitesimal limit. Theorem 5 .1 (Cheeger-Kleiner differentiabilit y theorem [25, 2 8]). F or almost every x ∈ U ther e exists a me asur e Σ x f on Cut( H ) such that for al l y , z ∈ H we have lim θ → 0 k f ( xδ θ ( y )) − f ( xδ θ ( z )) k 1 θ = Z Cut( H ) | 1 E ( y ) − 1 E ( z ) | d Σ x f ( E ) . (19) Mor e over, the me asure Σ x f is su pp orte d on affine half-sp ac es whose b ou n dary is a vertic al plane, i.e., a plane which isn ’t of the form H g for some g ∈ H (e quivalently, an inverse image, with r esp e ct to the ortho gonal pr oje ct ion fr om R 3 onto R × R ×{ 0 } , of a line in R × R × { 0 } ). Theorem 5.1 is incompatible with f b eing bi-Lipschitz, since the right hand side of (19) v anishes when y , z lie on the same coset of the cent er of H , while if f is bi-Lipschitz the left hand side of (1 9) is at least a constant mu ltiple o f d H ( y , z ). 5.4. Compression b ounds for L 1 em beddings of t he H eisen- b erg group. Theorem 1.1 and Theo r em 1.2 ar e b oth a consequence o f the following result from [29]: Theorem 5.2 (Quantitativ e ce n tral colla pse [2 9]). Ther e ex ists a universal c on- stant c ∈ (0 , 1) such t hat for every p ∈ H , every 1 -Lipschitz f : B ( p, 1) → L 1 , and every ε ∈  0 , 1 4  , t her e exists r > ε s u ch t hat with r esp e ct to Haar me a- sur e, for at le ast half of the p oints x ∈ B ( p, 1 / 2 ) , at le ast half of the p oints ( x 1 , x 2 ) ∈ B ( x, r ) × B ( x, r ) which lie on the same c oset of the c enter satisfy: k f ( x 1 ) − f ( x 2 ) k 1 6 d H ( x 1 , x 2 ) (log(1 / ε )) c . It isn’t difficult to see that Theor em 5.2 implies Theorem 1 .1 and Theo rem 1.2. F or ex ample, in the setting of Theor em 1.1 we are given a bi-Lipschitz embedding f : { 1 , . . . , n } 3 → L 1 , and using either the ge ner al extensio n theorem of [51] or a partition of unity arg umen t, we can extend f to a Lipschitz (with resp ect to d H ) mapping ¯ f : [1 , n ] 3 → L 1 , who se Lipschitz co nstant is at most a constant multiple of the Lipschitz co nstant of f . Theorem 5.2 (a fter r escaling by n ) pro duces a pair of po int s y , z ∈ [1 , n ] 3 of distance & √ n , who se distance is co nt racted under ¯ f 20 Assaf Naor by & (log n ) c . By rounding y , z to their nearest integer p oints in { 1 , . . . , n } 3 , w e conclude that f itself must hav e bi-Lipschitz distortion & (log n ) c . The deduction of T he o rem 1.2 from Theore m 5.2 is just a s simple; see [29]. Theorem 5.2 is a q uantitativ e version o f Theo rem 5.1, in the sense it gives a definite low er b ound on the macr oscopic sca le at which a given amount of co lla pse of cos ets of the center, as exhibited by the different iation result (19), o c c ur s. As explained in [2 9, Rem. 2.1], one cannot hop e in gener al to obtain r ate b ounds in differentiation r e sults suc h a s (19). Nevertheless, there are situations where “ q uan- titative differe n tiation re s ults” have b een successfully proved; impo r tant pr ecursor s of Theo rem 5.2 include the work o f Bourg ain [17], Jones [4 3], Matouˇ sek [57], and Bates, Johnson, Lindenstr a uss, P r eiss, Schec h tman [1 3]. Specifica lly , we should men tion that Bo urgain [17] obtained a low er bound on ε > 0 such that any em bed- ding of an ε -net in a unit ba ll of an n - dimens io nal normed space X into a nor med space Y ha s roughly the sa me distortion as the distortion req uired to embed all o f X in to Y , and Ma tou ˇ sek [5 7], in his study of em bedding s of trees into uniformly conv ex spac es, obtained quantitativ e b ounds on the sca le a t which “metr ic differ- ent iation” is almost a chieved, i.e., a scale at which discrete g eo desics are mapp ed by a Lipschitz function to “almost geo desic s”. These earlier results a re in the spirit of Theorem 5 .2, though the pro of of Theorem 5.2 in [29] is s ubstantially more inv olv ed. W e shall now say a few w ords on the pro of of Theo rem 5.2; for lac k of space this will hav e to b e a roug h sketc h, so we refer to [29] for more deta ils, as well as to the somewhat different presentation in [30]. Cheeger and Kleiner obtained tw o different pro ofs of Theor em 5.1. T he fir st pro of [25] s tarted with the imp ortant obser v ation that the fact that f is Lipschitz forces the cut measure Σ f to be supp orted on sets w ith additional regular it y , namely sets of finite p erimeter . Mo reov er, ther e is a definite bo und on the total p erimeter : R Cut( U ) PER( E , B ( p, 1 )) d Σ f ( E ) . 1, where PER( E , B ( p, 1)) denotes the p e rimeter of E in the ball B ( p, 1) (we refer to the bo o k [2], and the detailed explanation in [25, 29] for mo re information on these notions). Theo rem 5.2 is then prov ed in [25] via a n app eal to r esults [35, 36] on the infinitesimal str ucture of sets of finite p er imeter in H . A different pro of of Theorem 5 .2 was found in [28]. It is base d on the notion o f metric differ entiation , which is used in [28] to reduce the problem to mappings f : H → L 1 for which the cut mea sure is supp orted on monotone sets , i.e., sets E ⊆ H such that for every horizontal line L , up to a se t of measure zero , b oth L ∩ E and L ∩ ( H \ E ) are either empt y or subr ays of L . A non-trivial classifica tion of monotone sets is then prov ed in [28]: such sets a r e up to measure zero half-spac e s. This second pr o of o f Theore m 5.2 av oids completely the use of per imeter bo unds. Nevertheless, the star ting point of the proof of Theorem 5.2 can be viewed as a h ybrid a rgument, which incorp orates b oth per imeter bounds, and a new classi- fication o f almost monotone sets. The quantitativ e setting of Theo rem 5.2 leads to issues that do no t have analogues in the no n- quantitativ e pro o fs (e.g., the approx- imate classificatio n results of “almost” mo notone sets in balls c annot b e simply that such sets are clos e to ha lf-spaces in the entire ball; s ee [29, Example 9.1 ]). In order to pro ceed we need to qua n tify the extent to which a set E ⊆ B ( x, r ) Em b eddings of the H eisen b erg group 21 is monotone. F or a horizontal line L ⊆ H define the non-conv exity NC B ( x,r ) ( E , L ) of ( E , L ) on B ( x, r ) as the infimum of R L ∩ B ( x, r )   1 I − 1 E ∩ L ∩ B r ( x )   d H 1 L ov er all sub-interv als I ⊆ L ∩ B r ( x ). Here H 1 L is the 1-dimensio nal Hausdorff meas ure on L (induced fr om the metric d H ). The non- monotonicity of ( E , L ) on B ( x, r ) is defined to b e NM B ( x,r ) ( E , L ) def = NC B ( x,r ) ( E , L ) + NC B ( x,r ) ( H \ E , L ). The total non-monotonicity of E on B ( x, r ) is defined as : NM B ( x,r ) ( E ) def = 1 r 4 Z lines( B ( x,r )) NM B ( x,r ) ( E , L ) d N ( L ) , where lines( U ) denotes the set of horizontal lines in H which intersect U , and N is the left inv a r iant measur e on lines( H ), norma liz ed so that the mea s ure of lines( B ( e, 1)) is 1. The following stability result for monotone sets constitutes the bulk of [29]: Theorem 5.3. Ther e ex ists a universal c onst ant a > 0 s u ch that if a me asur able set E ⊆ B ( x, r ) satisfies NM B ( x,r ) ( E ) 6 ε a then ther e ex ists a half-sp ac e P such that L 3 (( E ∩ B εr ( x )) △P ) L 3 ( B εr ( x )) < ε 1 / 3 . Perimeter b ounds are used in [29, 3 0] for tw o purp oses. The first is finding a c ontrolled scale r s uch that at most lo ca tions, apa rt from a certain collection of cuts, the mass of Σ f is supp orted o n subsets which s atisfy the as sumption of Theorem 5 .3 (see [30, Sec. 9]). B ut, the excluded cuts may hav e infinite measure with resp ect to Σ f . Nonetheless, using p erimeter b ounds once more, together with the iso p erimetric inequality in H (see [60, 21]), it is shown that their contribution to the metric is negligibly sma ll (se e [30, Sec. 8]). By Theorem 5.3, it remains to deal with the situation where a ll the cuts in the suppo rt of Σ f are close to half-spaces: note that w e are not claiming in Theorem 5.3 that the half-s pace is vertical. Nevertheless, a s imple g eometric argument shows that even in the ca se of cut measur es that ar e suppo rted on genera l (a lmost) half- spaces, the mapping f must significantly distort s ome distances. The key p oint here is that if the cut meas ure is a c tually supp orted on half spaces, then it follows (after the fa c t) that for every affine line L , if x 1 , x 2 , x 3 ∈ L and x 2 lies betw een x 1 and x 3 then k f ( x 1 ) − f ( x 3 ) k 1 = k f ( x 1 ) − f ( x 2 ) k 1 + k f ( x 2 ) − f ( x 3 ) k 1 . 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