A geometric study of Wasserstein spaces: Euclidean spaces

A GEOMETRIC STUD Y OF W ASSERSTEIN SP ACES: EUCLIDEAN SP ACES by Beno ˆ ıt Klo ec kner A bstr a ct . — In this articl e we consider W asserstein spaces (with quadratic trans- portation cost) as intrinsic metric spaces. W e are in terested in usual geometric prop- erties: curv at ure, rank and isometry group, mostly in the case of Euclidean spaces. Our most striking result i s that the W asserstein space of the li ne admits “exotic” isometries, which do not preserve the shap e of measures. Con ten ts 1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 2. The W asser stein space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 3. Geo desics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 4. Curv ature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 5. Isometries: the cas e o f the line . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 6. Isometries: the higher -dimensional case . . . . . . . . . . . . . . . . . . . . . . . . 18 7. Ranks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 8. Op en problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 1. In tro duction The co ncept o f o ptimal tra nspo r tation r ecent ly r aised a growing interest in links with the ge o metry o f metric spa ces. In particular the L 2 W asserstein space W 2 ( X ) hav e b een used by V on Renesse and Sturm [ 15 ], Sturm [ 1 7 ] and Lo tt and Villani [ 12 ] to define cer ta in curv ature conditions on a metric s pace X . Ma n y useful pro per ties are inherited from X by W 2 ( X ) (separa bilit y , completeness, geo desicness, so me non- negative curv ature conditions) while s ome other are not, like lo ca l compacity . 2000 Mathematics Subje ct Classific at ion . — 54E70, 28A33. Key wor ds and phr ases . — W asserstein distance, optimal transp ortation, isometries, rank. 2 BENO ˆ IT KLOECKNER In this pap er, w e study the geometry of W a sserstein spaces as in trinsic spa c e s. W e are interested, for example, in the iso metr y g roup of W 2 ( X ), in its curv ature and in its rank (the greatest p ossible dimensio n of a E uclidean space that embeds in it). In the case of the W a sserstein space of a Riemannian manifold, itself see n as a n infinite- dimensional Riemannian manifold, the Riema nnian connection and curv ature ha ve bee n co mputed by Lott [ 13 ]. See also [ 18 ] where T ak atsu studies the subspace of Gaussian measures in W 2 ( R n ), and [ 1 ] wher e Am brosio and Gigli are in terested in the second order ana ly sis on W 2 ( R n ), in par ticular its parallel tr a nspo rt. The W asserstein spa ce W 2 ( X ) contains a copy of X , the image of the isometric embedding E : X → W 2 ( X ) x 7→ δ x where δ x is the Dirac mass a t x . Mo r eov er, given an isometry ϕ of X one defines an isometry ϕ # of W 2 ( X ) b y ϕ # ( µ )( A ) = µ ( ϕ − 1 ( A )). W e th us get an embedding # : Isom X → Isom W 2 ( X ) These tw o elemen tary facts connect the g eometry of W 2 ( X ) to that o f X . One could expect that # is onto, i.e. that all isometries o f W 2 ( X ) are induced by those of X itself. Elements of #(Isom X ) are ca lled trivial isometries. Let us int ro duce a w eaker prop erty: a self-map Φ of W 2 ( X ) is said to pr eserve shap es if for all µ ∈ W 2 ( X ), there is an isometr y ϕ of X (that dep ends upo n µ ) such that Φ( µ ) = ϕ # ( µ ). An isometry that do es not preser ve shapes is said to b e ex otic . Our ma in result is the s urprising fa c t that W 2 ( R ) admits exotic iso metries. More precisely we prov e the fo llowing. The or em 1. 1 . — The isometry gr oup of W 2 ( R ) is a semidir e ct pr o duct (1) Isom R ⋉ Isom R Both factors de c omp ose: Is om R = Z / 2 Z ⋉ R and the action defining the semidir e ct pr o duct (1) is simply given by the usu al action of the left Z / 2 Z factor on the right R factor, that is ( ε, v ) · ( η , t ) = ( η , εt ) wher e Z / 2 Z is identifie d with {± 1 } . In (1) , the left factor is the image of # and the right factor c onsist in al l isometries that fix p o intwise the s et of Dir ac masses. In the de c omp osition of the latter, the Z / 2 Z factor is gener ate d by a non- trivial involution that pr eserves shap es, while the R factor is a flow of exotic isometries. The main to ol we use is the explicit descr iption o f the geo desic b et ween tw o p oints µ 0 , µ 1 of W 2 ( R ) that follows from the fact that the unique o ptimal transpo r tation plan b etw een µ 0 and µ 1 is the non-decr easing rearr angement. It implies that most of the g eo de s ics in W 2 ( R ) are not complete, and we rely on this fact to give a metric characterization of Dirac masses and of linear combinations of tw o Dirac masses , among all p oints of W 2 ( X ). W e a lso use the fact that W 2 ( R ) has v anishing curv ature in the sens e o f Alexandrov. Let us describ e roughly the non-trivial isometries that fix p oint wise the set of Dirac masses. O n the one hand, the non-trivial isometry g enerating the Z / 2 Z factor W ASSERSTEIN SP A CE OF EUCLIDEAN SP A CES 3 is defined as follows: a mea sure µ is mapp ed to its symmetric with resp ect to its center of mass . O n the other hand, the exotic isometric flow tends to put all the ma ss on one side of the cen ter of gravit y (that must be pres erved), close to it, and to send a small bit of mas s far awa y on the other side (so that the W ass erstein distance to the center of mass is pr eserved). In par ticular, under this flow any measure µ co n verges weakly (but o f co urse not in W 2 ( R )) to δ x (where x is the center of mass of µ ), see Prop osition 5.4. The ca se of the line s eems very sp ecial. F o r ex ample, W 2 ( R n ) admits non-triv ial isometries but a ll of them prese r ve shape s . The or em 1. 2 . — If n > 2 , the isometry gr oup of W 2 ( R n ) is a semidir e ct pr o duct (2) Isom( R n ) ⋉ O( n ) wher e t he action of an element ψ ∈ Isom( R n ) on O( n ) is the c onjugacy by its line ar p art ~ ψ . The left factor is the image of # and e ach element in the r ight factor fixes al l Dir ac masses and pr eserves s hap es. The proo f relies on Theo rem 1.1, some elemen tary prop erties of L 2 optimal trans- po rtation in R n and Radon’s T he o rem [ 14 ]. W e s ee that the q uotien t Isom W 2 ( R n ) / Isom R n is compact if and o nly if n > 1. The hig her-dimensional Euclidea n space s are more r igid than the line fo r this pro ble m, and we e x pect mos t of the other metric spaces to be even mor e rigid in the sense tha t # is onto. Another consequence of the study of complete geo desics co ncerns the rank o f W 2 ( R n ). The or em 1. 3 . — Ther e is no isometric emb e d ding of R n +1 into W 2 ( R n ) . It is simple to prov e that despite Theo rem 1.3, large pieces of R n can be embedded int o W 2 ( R ), which has consequently infinite weak rank in a sense to be precised. As a consequence, we get for example: Pr op osition 1. 4 . — If X is any Polish ge o desic metric sp ac e that c ontains a c om- plete ge o desic, then W 2 ( X ) is n ot δ -hyp erb olic. This is no t surprising, since it is well-kno wn that the nega tive c urv a tur e a s sump- tions tend no t to b e inherited fr o m X b y its W asser stein space. An explicit example is co mputed in [ 2 ] (Exa mple 7.3.3 ); more generaly , if X contains a rhom bus (four distinct p oin ts x 1 , x 2 , x 3 , x 4 so that d ( x i , x i +1 ) is indep endent of the cyclic index i ) then W 2 ( X ) is not uniquely g eo desic, a nd in particular not CA T(0), even if X itself is strongly neg atively curved. Organization of the pap er. — Sections 2 to 4 colle c t some pr oper ties needed in the sequel. Theorem 1.1 is pr ov ed in Sec tion 5, Theore m 1.2 in Section 6. Section 7 is devoted to the ranks of W 2 ( R ) and W 2 ( R n ), and we end in Section 8 with so me op en questions. 4 BENO ˆ IT KLOECKNER Ac kno wledgemen ts. — I wish to thank all sp eakers of the workshop on optimal transp ortation held in the Institut F our ie r in Grenoble, esp ecially Nicolas Juillet with whom I had numerous discussion on W asser stein spaces, a nd its org anizer Herv ´ e P a jot. I am also indebted to Y ann Ollivier for advises and p ointing out s o me inaccur acies and mistakes in preliminary versions of this pap er. 2. The W asserstein space In this pr e liminary section we recall well-known ge ne r al facts on W 2 ( X ). One ca n refer to [ 19, 20 ] for further details and muc h mo re. Note that the denomina tio n “W asserstein s pace” is deba ted a nd historica lly inacc urate. How ev er, it is now the most common denomination a nd thus an o ccurrence of the self-applying theo rem of Arnol’d according to which a mathematical res ult or ob ject is us ually a ttributed to someone that had little to do with it. 2.1. Geo desi c s paces. — Let X b e a Polish ( i.e. co mplete and separable metric) space, and assume tha t X is geo desic, that is: b etw een t wo po in ts there is a r e ctifiable curve whose length is the distance b etw een the c onsidered po ints. Note tha t we only consider glob ally minimizing geo desics, a nd that a ge o des ic is always assumed to b e parametrized prop ortio nally to arc length. One defines the W asserstein s pace of X as the set W 2 ( X ) of Borel probability measures µ on X that satisfy Z X d 2 ( x 0 , x ) µ ( dx ) < + ∞ for some (hence all) po in t x 0 ∈ X , equipp e d by the distance d W defined by: d 2 W ( µ 0 , µ 1 ) = inf Z X × X d 2 ( x, y )Π( dxdy ) where the infimum is taken over a ll coupling Π of µ 0 , µ 1 . A coupling realizing this infim um is s aid to be optimal , and there alwa ys exists an optimal co upling. The idea behind this distance is link ed to the Monge - Kantoro vitch proble m: given a unit quantit y of go o ds distributed in X accor ding to µ 0 , what is the most economical wa y to dis pla ce them so that they end up distributed accor ding to µ 1 , when the cost to mov e a unit of go o d fro m x to y is given b y d 2 ( x, y )? The minimal cost is d 2 W ( µ 0 , µ 1 ) and a transp ortatio n plan achieving this minimum is an optima l coupling. An optimal coupling is said to b e deterministic if it can b e written under the form Π( dxdy ) = µ ( dx ) 1 [ y = T x ] where T : X → X is a measur a ble map and 1 [ A ] is 1 if A is sa tisfied and 0 other w is e. This means that the coupling do es no t split mass : all the mass a t p oint x is moved to the p oint T x . One usually write Π = (Id × T ) # µ . Of course, for Π to b e a coupling b etw een µ and ν , the rela tio n ν = T # µ m ust hold. Under the assumptions we put on X , the metric space W 2 ( X ) is itself Polish and geo desic. If moreover X is uniquely geo desic , then to each o ptimal coupling Π b e- t ween µ 0 and µ 1 is asso ciated a uniq ue g eode s ic in W 2 ( X ) in the following way . Let C ([0 , 1] , X ) b e the set o f contin uous curves [0 , 1] → X , let g : X × X → C ([0 , 1] , X ) be the applica tion that maps ( x, y ) to the co nstant s peed geo desic be t w een thes e W ASSERSTEIN SP A CE OF EUCLIDEAN SP A CES 5 po in ts, a nd for e ach t ∈ [0 , 1] let e ( t ) : C ([0 , 1] , X ) → X b e the map γ 7→ γ ( t ). Then t 7→ e ( t ) # g # Π is a geo desic b et w een µ 0 and µ 1 . Informally , this means that we cho ose randomly a couple ( x, y ) a ccording to the jo in t law Π, then take the time t of the geo desic g ( x, y ). This gives a r andom p oint in X , who se law is µ t , the time t of the geo desic in W 2 ( X ) asso cia ted to the optimal coupling Π. Moreov er, a ll geo desics ar e obtained that way . Note that for most spaces X , the optimal coupling is no t unique for all pairs of probability meas ures, and W 2 ( X ) is therefore no t uniquely geo desic even if X is. One of our g oal is to deter mine whether the Dirac measures ca n be detected inside W 2 ( X ) by purely geometric prop erties, so that we can link the isometr ies of W 2 ( X ) to those o f X . 2.2. The line. — Given the distribution function F : x 7→ µ (] − ∞ , x ]) of a pro babilit y measure µ , one defines its left-contin uous in verse: F − 1 : ]0 , 1 [ → R m 7→ sup { x ∈ R ; F ( x ) 6 m } that is a no n-decreasing, le ft-con tinu ous function; lim 0 F − 1 is the infimum of the suppo rt o f µ and lim 1 F − 1 its supremum. A discontin uit y of F − 1 happ ens for each int erv al that do es no t intersect the supp ort of µ , and F − 1 is consta n t on an in terv al for each atom of µ . m x Figure 1. Inv erse distribution function of a combination of three D irac masses Let µ 0 and µ 1 be tw o p oints of W 2 ( R ), and let F 0 , F 1 be their distribution functions. Then the dista nce b etw een µ 0 and µ 1 is given by (3) d 2 ( µ 0 , µ 1 ) = Z 1 0  F − 1 0 ( m ) − F − 1 1 ( m )  2 dm and there is a unique constant speed geo desic ( µ t ) t ∈ [0 , 1] , where µ t has a dis tr ibution function F t defined by (4) F − 1 t = (1 − t ) F − 1 0 + tF − 1 1 6 BENO ˆ IT KLOECKNER This means that the b est wa y to g o from µ 0 to µ 1 is simply to rear range increasingly the ma ss, a co nsequence of the conv exity of the cost function. F or example, if µ 0 and µ 1 are uniform measure s on [0 , 1] and [ ε, 1 + ε ], then the optimal coupling is deterministic given b y the translation x 7→ x + ε . That is: the b est way to go from µ 0 to µ 1 is to shift every bit of mass by ε . If the co st function where linear, it would be equiv a len t to leav e the mass on [ ε, 1] where it is and mov e the r emainder from [0 , ε ] to [1 , 1 + ε ]. If the cos t function where concave, then the latter solution w ould b e better than the for mer. m x Figure 2. A geo desic b etw een tw o atomic measures: the mass mov es with sp eed prop ortional to the length of the arrows. 2.3. Higher di mensional Euclide an s paces. — The Mong e-Kantorovic h prob- lem is far mor e in tricate in R n ( n > 2 ) than in R . The ma jor contributions o f Knott and Smith [ 11, 16 ] a nd Brenier [ 3, 4 ] g iv e a quite sa tisfactory characterization o f optimal couplings and their unicity when the tw o cons idered measures µ and ν ar e absolutely contin uous (with resp ect to the Leb esgue measure ). W e shall not give details of these works, for which we refer to [ 19, 20 ] again. Let us how ever co ns ider some toy cases, which will prove useful later on. Missing proo fs can b e found in [ 10 ], Section 2.1.2. W e consider R n endow ed with its ca nonical inner pr o duct and no rm, denoted by | · | . T r anslatio ns . — Let T v be the translation o f vector v a nd assume that ν = ( T v ) # µ . Then the unique optimal coupling b et ween µ and ν is deterministic, eq ua l to (Id × T v ) # µ , and therefor e d W ( µ, ν ) = | v | . This mea ns that the only mos t economic way to move the mass from µ to ν is to tr anslate each bit of mass by the v ector v . This is a quite intuit ive co nsequence of the conv exit y of the cost. In particular, the geo desic betw een µ and ν can be ex tended for all times t ∈ R . This ha pp ens only in this case as we s hall see later on. Dilations . — Let D λ x be the dilation of ce nter x a nd ratio λ a nd ass ume that ν = ( D λ x ) # µ . The n the unique optimal co upling b etw een µ a nd ν is deterministic, equal W ASSERSTEIN SP A CE OF EUCLIDEAN SP A CES 7 to (Id × D λ x ) # µ . In particular , d W ( µ, ν ) = | 1 − λ 2 | 1 2 d W ( µ, δ x ) As a consequence, the geo des ic b etw een µ and ν is unique and made of homo thetic of µ , and can be extended only to a semi-infinite int erv al: it cannot b e extended beyond δ x (unless µ is a Dirac mass itself ). Ortho go nal me a sur es . — Assume that µ and ν are supp orted o n o rthogonal affine subspaces V and W of R n . Then if Π is any coupling, as s uming 0 ∈ V ∩ W , we hav e Z R n × R n | x − y | 2 Π( dxdy ) = Z R n × R n ( | x | 2 + | y | 2 )Π( dxdy ) = Z V | x | 2 µ ( dx ) + Z W | y | 2 ν ( dy ) therefore the co st is the same whatever the coupling. Balanc e d c ombinations of two Dir a c masses . — Assume that µ = 1 / 2 δ x 0 + 1 / 2 δ y 0 and ν = 1 / 2 δ x 1 + 1 / 2 δ y 1 . A co upling b etw een µ and ν is ent irely deter mined by the amount m ∈ [0 , 1 / 2] of mass s en t from x 0 to x 1 . The cost of the co upling is 1 2 | x 1 − y 0 | 2 + 1 2 | x 0 − y 1 | 2 − 2 m ( y 0 − x 0 ) · ( y 1 − x 1 ) th us the optimal co upling is unique and deter minis tic if ( y 0 − x 0 ) · ( y 1 − x 1 ) 6 = 0 , given b y the map ( x 0 , y 0 ) 7→ ( x 1 , y 1 ) if ( y 0 − x 0 ) · ( y 1 − x 1 ) > 0 and by the map ( x 0 , y 0 ) 7→ ( y 1 , x 1 ) if ( y 0 − x 0 ) · ( y 1 − x 1 ) < 0 (figure 3 ). Of cours e if ( y 0 − x 0 ) · ( y 1 − x 1 ) = 0, then all coupling hav e the same cost and a re therefore optimal. Figure 3. Optimal coupling b etw een balanced com binations of tw o Dirac masses. Contin uous arrows represent the vectors y 0 − x 0 and y 1 − x 1 while dashed arrows represent th e optimal coupling. If the combinations ar e not balanced (the ma s s is not equally split betw een the tw o po in t of the support), then the optimal c oupling is easy to deduce from the preceding computation. F or example if ( y 0 − x 0 ) · ( y 1 − x 1 ) > 0 then as muc h mass as p ossible m ust b e sent from x 0 to x 1 , and this deter mines the optimal co upling. This ex ample has a muc h more general impa ct than it might seem: it can be generalized to the following (very) sp ecial case of the cyclic al monotonicity (see for example [ 20 ], Chapter 5 ) which will prov e use ful in the seq ue l. L emma 2. 1 . — If Π is an optimal c o upling b etwe en any two pr ob ability me a sur es on R n , then ( y 0 − x 0 ) · ( y 1 − x 1 ) > 0 8 BENO ˆ IT KLOECKNER holds whenever ( x 0 , x 1 ) and ( y 0 , y 1 ) ar e in the s u pp ort of Π . 2.4. Spaces of nonp os i tiv e curv ature. — In this pap er we shall consider tw o cur- v a ture conditions. The fir st one is a neg ative curv ature co ndition, the δ -hyperb olicity int ro duced by Gro mov (se e for example [ 5 ]). A geo desic space is said to b e δ - hyperb olic (where δ is a non-neg ative num ber) if in any tria ngle, any p oint of any o f the sides is at distance at most δ fro m one of the other tw o sides. F or example, the real hyperb olic space is δ -hyperb olic (the v alue o f δ dep ending on the v alue o f the curv ature), a tree is 0-hyper bolic and the euclidean spaces of dimensio n at leas t 2 are not δ -hyperb olic for a n y δ . The se cond condition is the c la ssical non-p ositive sectional curv ature condition CA T(0), detailed in Se c tio n 4 , tha t roughly means that triangles a re thinner in X than in the euclidean plane. Euclidean spaces , any Riemannian manifold having non-p ositive sectiona l curv a tur e are examples of lo cally C A T(0) spaces. A geo desic CA T(0) Polish s pace X is also ca lled a Hadamar d sp ac e . A Hadama rd space is uniquely geo desic, a nd a dmits a natural bo undary at infinity . The feature that in terests us mo st is the following cla ssical result: if X is a Hada ma rd space, given µ ∈ W 2 ( X ) there is a unique p oint x 0 ∈ X , called the center o f mass of µ , that minimizes the quantit y R X d 2 ( x 0 , x ) µ ( dx ). If X is R n endow ed with the canonica l scalar product, then the cent er of ma ss is of course R R n xµ ( dx ) but in the general case, the la ck of an affine s tr ucture on X preven ts the use of such a formula. W e thus get a map P : W 2 ( X ) → X that maps any L 2 probability measur e to its center o f mass. Ob viously , P is a left inv erse to E and one can hop e to use this map to link clos e r the geometr y o f W 2 ( X ) to that o f X . That’s why our questions, unlike most of the c la ssical ones in optimal transp ortatio n, might behave mor e nicely when the curv ature is non-p ositive than when it is non-negative. 3. Geo desi cs The con ten t of this section, although difficult to lo cate in the bibliography , is part of the folklore a nd do es not pretend to or iginality . W e giv e pro o fs for the sak e of completeness. 3.1. Case of the line. — W e now consider the geodes ics of W 2 ( R ). Our first go al is to determine on which max imal in terv al they can b e extended. Maximal exten sion . — Let µ 0 , µ 1 be tw o po in ts of W 2 ( R ) and F 0 , F 1 their distribution functions. Let ( µ t ) t ∈ [0 , 1] be the g eo desic b e tween µ 0 and µ 1 . Since W 2 ( R ) is uniquely geo desic, there is a unique maximal int erv al o n which γ can be extended into a geo desic, denoted by I ( µ 0 , µ 1 ). L emma 3. 1 . — One has I ( µ 0 , µ 1 ) = { t ∈ R ; F − 1 t is non-de cr e asing } wher e F − 1 t is define d by the formula (4 ) . It is a close d int erval. If one of its b ound t 0 is fi nite, then µ t 0 do es not have b ounde d density with r esp e ct to the Le b esgue me asur e. W ASSERSTEIN SP A CE OF EUCLIDEAN SP A CES 9 Pr o of . — Any non-decr easing left co ntin uous function is the inv erse dis tribution func- tion of some proba bility mea sure. If such a function is obtained by an affine combi- nation o f proba bilities b e longing to W 2 ( R ), then its probability mea sure b elongs to W 2 ( R ) to o. Moreov er, an affine co m bination of tw o left co n tin uous function is left co n tin uous, so that I ( µ 0 , µ 1 ) = { t ∈ R ; F − 1 t is non-decr easing } . The fact tha t I ( µ 0 , µ 1 ) is closed follows from the stability of non- de c r easing func- tions under p oint wise conv ergence. If the minimal slop e inf  F − 1 t ( m ) − F − 1 t ( m ′ ) m − m ′ ; 0 < m < m ′ < 1  is p ositive for some t , then it s ta ys p ositive in a neighborho o d of t . Thus, a finite b ound of I ( µ 0 , µ 1 ) must ha ve zero minimal slop e, and cannot hav e a b ounded densit y . A geo desic is said to b e c omplete if it is defined for a ll times. W e also consider geo desic r ays, defined on an interv al [0 , T ] or [0 , + ∞ [ (in the latter ca se w e say that the ray is complete), and geo desic s egment s, defined on a c lo sed int erv al. It is easy to deduce a num ber o f consequences from Lemma 3.1. Pr op osition 3. 2 . — In W 2 ( R ) : (1) any ge o desic r ay issue d fr om a Dir ac mass c an b e extende d to a c omplete r a y, (2) no ge o desic ra y issue d fr om a Dir ac mass c an b e exten de d for ne gative times, exc ept if al l of its p oints ar e Dir ac masses, (3) up to normalizing the sp e e d, t he only c omplete ge o desics ar e those obtaine d by tr anslating a p oint of W 2 ( R ) : µ t ( A ) = µ 0 ( A − t ) , Pr o of . — The inv erse distribution function of a Dirac mass δ x is the constant function F − 1 0 with v alue x . Since it slop es F − 1 0 ( m ) − F − 1 0 ( m ′ ) m − m ′ 0 < m < m ′ < 1 are all zero, for all p ositive times t the functions F − 1 t defined by formula (4) for any non-decr easing F − 1 1 are non-decreas ing . Howev er, for t < 0 the F − 1 t are not non-decreas ing if F − 1 1 is not consta n t, w e th us get (1) and (2). Consider a point µ 0 of W 2 ( R ) defined b y an inverse distributio n function F − 1 0 , and consider a complete geo desic ( µ t ) issued from µ 0 . Let F − 1 t be the inverse distr ibution function of µ t . Then, since µ t is defined for all times t > 0, the slop es of F − 1 1 m ust be grea ter than those of F − 1 0 : F − 1 0 ( m ) − F − 1 0 ( m ′ ) 6 F − 1 1 ( m ) − F − 1 1 ( m ′ ) ∀ m < m ′ otherwise, when t increases, some s lope of F − 1 t will decr ease linear ly in t , thus b e- coming negative in finite time. 10 BENO ˆ IT KLOECKNER But since µ t is also defined for all t < 0, the s lopes of F − 1 1 m ust b e lesse r than those o f F − 1 0 . They ar e therefore equa l, and the tw o inverse distribution function are equal up to a n additive co nstant . The geo desic µ t is the tra nslation o f µ 0 and we prov ed (3 ). Convex hul ls of total ly atomic me asur es . — Define in W 2 ( R ) the following sets: ∆ 1 = { δ x ; x ∈ R } ∆ n = { n X i =1 a i δ x i ; x i ∈ R , P a i = 1 } ∆ ′ n +1 = ∆ n +1 \ ∆ n Recall that if X is a Polish g eo des ic space and C is a subset of X , one says that C is convex if every geo desic segment whos e endpo in ts are in C lies en tirely in C . The conv ex hull o f a subset Y is the least co n vex set C ( Y ) that co n tains Y . It is well defined s ince the intersection o f tw o conv ex sets is a conv ex set, and is equa l to ∪ n ∈ N Y n where Y 0 = Y and Y n +1 is obtained by adding to Y n all p oints lying on a geo desic whose endp oint s ar e in Y n . Since ∆ 1 is the image o f the iso metric embedding E : R → W 2 ( R ), it is a convex set. This is not the ca se of ∆ n is n > 1. In fact, we ha ve the following. Pr op osition 3. 3 . — If n > 1 , any p oint µ of ∆ n +1 lies on a ge o desic s e gment with endp oints in ∆ n . Mor e over, the endp oints c an b e chosen with the same c enter of mass than that of µ . Pr o of . — If µ ∈ ∆ n the result is obvious. Assume µ = P a i δ x i is in ∆ ′ n +1 . W e ca n assume further that x 1 < x 2 < · · · < x n +1 . Consider the meas ures µ − 1 = X i 1 . Pr o of . — F ollows from Prop ositio n 3.3 since the set of totally atomic measures S n ∆ n is dense in W 2 ( R ). W ASSERSTEIN SP A CE OF EUCLIDEAN SP A CES 11 3.2. Complete geo de s ics in higher dimens ion. — In R n , the optimal coupling and thus the geo desics are not as explicit as in the case of the line. It is howev er po ssible to determine which geo desic can b e ex tended to all times in R . L emma 3. 5 . — L et µ = ( µ t ) t ∈ I b e a ge o desic in W 2 ( R n ) asso cia te d t o an optimal c o upling Π b etwe en µ 0 and µ 1 . Then for al l t imes r and s in I and al l p air of p oints ( x 0 , x 1 ) , ( y 0 , y 1 ) in the supp ort of Π , the fol lowing hold: | u | 2 + ( r + s ) u · v + rs | v | 2 > 0 wher e u = y 0 − x 0 and v = y 1 − x 1 − ( y 0 − x 0 ) . Pr o of . — Let us in tro duce the following notations: for all pair of p oints a 0 , a 1 ∈ R n , a t = (1 − t ) a 0 + ta 1 and Π r,s is the law of the r andom v ariable ( X r , X s ) where ( X 0 , X 1 ) is any random v ariable of law Π. As we alr eady said, Π r,s is an optimal co upling o f µ r , µ s whose corr esp o nding geo desic is the restrictio n o f ( µ t ) to [ r, s ]. Since Π r,s is optimal, ac cording to the cyc lical monotonicity (see Lemma 2.1) one has ( y r − x r ) · ( y s − x s ) > 0. But w ith the ab ov e nota tio ns, o ne ha s y r − x r = u + rv and y s − x s = u + sv , a nd we get the desired ineq ua lit y . Let us show why this Lemma implies that the o nly complete geo desics are those obtained by translation. There are immediate cons equences on the ra nk of W 2 ( R n ), see Theorem 1.3 and Section 7. Pr op osition 3. 6 . — L et µ = ( µ t ) t ∈ R b e a ge o d esic in W 2 ( R n ) define d for al l times. Then ther e is a ve ctor u such that µ t = ( T tu ) # µ 0 . This result holds even if n = 1, as s tated in Prop osition 3.2. Pr o of . — It is sufficient to find a u such that µ 1 = ( T u ) # µ 0 , since then there is o nly one geo desic from µ 0 to µ 1 . Consider any pair o f p oints ( x 0 , x 1 ) , ( y 0 , y 1 ) in the supp o rt o f the coupling Π b e- t ween µ 0 and µ 1 that defines the restriction of µ to [0 , 1]. Define u = y 0 − x 0 and v = y 1 − x 1 − ( y 0 − x 0 ). If v 6 = 0, then there ar e rea l num ber s r < s such that | u | 2 + ( r + s ) u · v + r s | v | 2 < 0. Then the coupling Π r,s betw een µ r and µ s that defines the restr ic tion of µ to [ r, s ], defined as ab ov e, cannot b e optimal. This is a contradiction with the assumption that µ is a geo desic. Therefore, for all ( x 0 , x 1 ) , ( y 0 , y 1 ) in the suppo rt of Π one ha s y 0 − x 0 = y 1 − x 1 . This amounts to say th at Π is deterministic, giv en b y a translation of vector u = y 0 − x 0 . 4. Curv ature Once aga in, this section mainly co llects some facts that are alr eady well-known but shall b e used o n the sequel. More details on the (sectional) curv a ture of metric spaces are a v a ila ble for example in [ 6 ] or [ 9 ]. W e sha ll consider the curv ature of W 2 ( R ), in the sense of Alexa ndrov. Given an y three p oints x, y , z in a geo desic metr ic space X , there is up to congruenc e 12 BENO ˆ IT KLOECKNER a unique compariso n triang le x ′ , y ′ , z ′ in R 2 , that is a triangle that satisfies d ( x, y ) = d ( x ′ , y ′ ), d ( y , z ) = d ( y ′ , z ′ ), and d ( z , x ) = d ( z ′ , x ′ ). One says that X has non-p ositive curvatur e (in the sense o f Alexa ndrov), o r is CA T(0), if for all x, y , z the distances b et ween tw o p oints on sides of this triang le is lesser than or equa l to the distance betw een the corresp onding points in the compar- ison triangle, se e figure 4. Equiv alen tly , X is CA T(0) if for a n y triangle x, y , z , any g e o des ic γ suc h that γ (0) = x a nd γ (1) = y , and any t ∈ [0 , 1], the following inequality holds: (5) d 2 ( y , γ ( t )) 6 (1 − t ) d 2 ( y , γ (0)) + td 2 ( y , γ (1)) − t (1 − t ) tℓ ( γ ) 2 where ℓ ( γ ) denotes the length of γ , that is d ( x, z ). x y z γ ( t ) x ′ y ′ z ′ Figure 4. The CA T(0) inequality: t he dashed segment is shorter in the triangle xy z than in the comparison triangle on the right. One says that X has vanishing cu rvatu r e if equality ho lds for all x, y , z , γ , t : (6) d 2 ( y , γ ( t )) = (1 − t ) d 2 ( y , γ (0)) + td 2 ( y , γ (1)) − t (1 − t ) tℓ ( γ ) 2 This is equiv alent to the condition that for a n y triangle x, y , z in X and any point γ ( t ) on any geo desic segment b e t w een x a nd z , the distance b et ween y and γ ( t ) is e qual to the c orresp onding distance in the compariso n tria ngle. Pr op osition 4. 1 . — The sp a c e W 2 ( R ) has vanishing Alexandr ov cu r vatu r e . Pr o of . — It follows from the ex pression (3) o f the distance in W 2 ( R ): if we denote by A, B , C the inv erse distribution functions of the three considered p oints x, y , z ∈ W 2 ( R ), we g et: d 2 ( y , γ ( t )) = Z 1 0 ( B − (1 − t ) A − tC ) 2 = Z 1 0  (1 − t ) 2 ( B − A ) 2 + t 2 ( B − C ) 2 +2 t (1 − t )( B − A )( B − C )  W ASSERSTEIN SP A CE OF EUCLIDEAN SP A CES 13 and using that (1 − t ) 2 = (1 − t ) − t (1 − t ) and t 2 = t − t (1 − t ), d 2 ( y , γ ( t )) = (1 − t ) d 2 ( y , x ) + td 2 ( y , z ) − t (1 − t ) Z 1 0 h ( B − A ) 2 +( B − C ) 2 − 2( B − A )( B − C ) i = (1 − t ) d 2 ( y , x ) + td 2 ( y , z ) − t (1 − t ) d 2 ( x, z ) . W e shall use the v anishing curv ature of W 2 ( R ) b y mea ns of the following result, where all subs e ts o f X ar e ass umed to be endow ed with the induced metric (that need therefore not b e inner). Pr op osition 4. 2 . — L et X b e a Polish uniquely ge o desic sp ac e with vanishing cur- vatur e. If Y is a subset of X and C ( Y ) is t he c onvex hul l of Y , then any isometry of Y c an b e extende d into an isometry of C ( Y ) . Pr o of . — Let ϕ : Y → Y the isometry to b e extended. Let x, y b e any p oints lying each on one geo desic segment γ , τ : [0 , 1] → X whos e endp oints a re in Y . Co nsider the unique g eo desics γ ′ , τ ′ that s a tisfy γ ′ (0) = ϕ ( γ (0)), γ ′ (1) = ϕ ( γ (1)) τ ′ (0) = ϕ ( τ (0)), τ ′ (1) = ϕ ( τ (1)) and the p oints x ′ , y ′ lying on them so that d ( x ′ , γ ′ (0)) = d ( x, γ (0)), d ( x ′ , γ ′ (1)) = d ( x, γ (1 )), and the same fo r y ′ . This ma kes sense since, ϕ b eing an isometry on Y , γ ′ has the length of γ and τ ′ that of τ . W e shall prove that d ( x ′ , y ′ ) = d ( x, y ). The v anishing of curv a ture implies that d ( x ′ , τ ′ (0)) = d ( x, τ (0)): the triangles γ (0) , γ (1) , τ (0) and γ ′ (0) , γ ′ (1) , τ ′ (0) have the sa me compar ison tria ngle. Similarly d ( x ′ , τ ′ (1)) = d ( x, τ (1)). No w x, τ (0) , τ (1) and x ′ , τ ′ (0) , τ ′ (1) hav e the same compa r- ison triangle, a nd the v a nis hing curv a ture assumption implies d ( x ′ , y ′ ) = d ( x, y ). x ′ γ ′ (1) y ′ x γ (1) y τ (1) τ ′ (0) τ ′ (1) γ (0) γ ′ (0) τ (0) Figure 5. All triangles b eing flat, the distance is the same b etw een x ′ and y ′ and b etw een x and y . In particula r, if x = y then x ′ = y ′ . W e can thus extend ϕ to the union of geo desic segments whose endp oints are in Y b y mapping any such x to the corres ponding x ′ . This is well-defined, and an isometry . Repe ating this op eratio n we can extend ϕ in to an isometry of C ( Y ). But X being complete, the contin uous extension of ϕ to C ( Y ) is well-defined and an isometry . 14 BENO ˆ IT KLOECKNER Note that the s ame result holds with the same pro of whe n the curv ature is constant but non-zero . Higher dimensional c ase . — P rop osition 4.1 does not hold in W 2 ( R n ). In effect, there are pairs of geo desics that meet at b oth endp oints (take measures whose supp ort lie on orthog onal subspaces of R n ). T ak ing a third po in t in one of the tw o geo desics, one gets a triangle in W 2 ( R n ) who s e co mparison tr iangle has its three vertices on a line. This implies that W 2 ( R n ) is no t CA T(0 ). The situation is in fact worse than that: in any neighborho o d U of an y p oint of W 2 ( R n ) one can find t wo differen t geo desics that meet a t their endp oints. O ne ca n say that this space ha s p ositive sectional curv ature at arbitrar ily small scales. 5. Isometrie s: the case of the line 5.1. Existence and uni cit y of the non-trivial isome tric flow. — In this sec- tion, we prov e Theor em 1.1. Let us start with the following c o nsequence of P r op osition 3.2. L emma 5. 1 . — An isometry of W 2 ( R ) must glob ally pr eserve the sets ∆ 1 and ∆ 2 . Pr o of . — W e sha ll exhibit some ge o metric prop erties tha t characterize the po in ts of ∆ 1 and ∆ 2 and must be pr eserved by isometries. First, according to P rop osition 3.2, the p oint s µ ∈ ∆ 1 are the o nly o nes to sa tisfy: every maximal geo desic ray starting at µ is complete. Since an isometry m ust map a geo desic (ray , segment) to another, this pr op ert y is prese rved by isometries of W 2 ( R ). Second, let us prov e that the p oint µ ∈ ∆ 2 are the only o nes that satisfy: any geo desic µ t such tha t µ = µ 0 and that can be extended to a maximal int erv al [ T , + ∞ ) with −∞ < T < 0 , has its endp oint µ T in ∆ 1 . This prop erty is obviously s atisfied by po in ts of ∆ 1 . It is also satisfied by every po in ts of ∆ ′ 2 . Indeed, denote b y F t the dis tribution function of µ t and write µ = aδ x + (1 − a ) δ y where x < y . Then if µ 1 do es no t write µ 1 = aδ x 1 + (1 − a ) δ y 1 with x 1 < y 1 , either – there a re tw o reals b, c s uch that a < b < c < 1 and F − 1 1 ( b ) < F − 1 1 ( c ), – there a re tw o reals b, c s uch that 0 < b < c < a and F − 1 1 ( b ) < F − 1 1 ( c ), or – µ 1 is a Dira c mass. In the first t wo cases, µ t is not defined for t < 0, and in the third one, it is not defined for t > 1. If µ 1 do es wr ite µ 1 = a δ x 1 + (1 − a ) δ y 1 , then either | y 1 − x 1 | = | y − x | and µ t is defined for a ll t , or | y 1 − x 1 | < | y − x | a nd µ t is only defined until a finite po sitive time, or | y 1 − x 1 | > | y − x | and µ t is defined from a finite negative time T where µ T ∈ ∆ 1 . Now if µ / ∈ ∆ 2 , its inv erse distribution function F − 1 takes three different v alues at some p o in ts m 1 < m 2 < m 3 . Consider the g eo desic b etw een µ and the measur e µ ′ whose in v erse distributio n function F ′− 1 coincide w ith F − 1 on [ m 1 , m 2 ] but is defined by F ′− 1 ( m ) − F − 1 ( m 2 ) = 2 ( F − 1 ( m ) − F − 1 ( m 2 )) W ASSERSTEIN SP A CE OF EUCLIDEAN SP A CES 15 m x m 1 m 3 m 2 Figure 6. The geodesic b etw een these inv erse distributions is defined for negative times, more precisely until it reaches the d ashed line. on [ m 2 , 1) (see figure 6). Then this geodes ic is defined for all po s itiv e times, but stops at some nonpo sitive time T . Since F − 1 takes different v alues at m 2 and m 3 , one can extend the geo desic for small nega tiv e times and T < 0. But the inv erse distribution function o f the endp oint µ T m ust tak e the same v alues than that of µ in m 1 and m 2 , th us µ T / ∈ ∆ 1 . Now w e consider isometries of ∆ 2 , to which all isometries of W 2 ( R ) shall b e reduce d. An y p o in t µ ∈ ∆ ′ 2 writes under the form µ = µ ( x, σ, p ) = e − p e − p + e p δ x − σ e p + e p e − p + e p δ x + σe − p where x is its center of mass, σ is the distance betw een µ and its center o f mass, and p is any re a l n umber. In probabilistic terms, if µ is the law of a r andom v ariable then x is its exp ected v alue and σ 2 its v ariance. L emma 5. 2 . — An isometry of W 2 ( R ) that fixes e ac h p oint of ∆ 1 must r estrict to ∆ ′ 2 to a map of the form: Φ( ϕ ) = µ ( x, σ, p ) 7→ µ ( x, σ, ϕ ( p )) for s ome ϕ ∈ Iso m( R ) . Any such map is an isometry of ∆ 2 . Pr o of . — Let Φ be an isometry o f W 2 ( R ) that fixes ea c h po in t of ∆ 1 . A computation gives the following expre s sion for the distance b etw een t w o measures in ∆ ′ 2 : d 2 W ( µ ( x, σ, p ) , µ ( y , ρ, q )) = ( x − y ) 2 + σ 2 + ρ 2 − 2 σ ρe | p − q | Since Φ is an isometry , it preserves the center o f mass and v ariance. The preceding expression shows that it must pr e serve the euclidean distance b etw een p and q for any tw o mea sures µ ( x, σ, e ) , µ ( y , ρ, f ), and that this conditio n is sufficient to mak e Φ an isometry of ∆ 2 . L emma 5. 3 . — L et ψ : x → εx + v and ϕ : p → η p + t b e isometries of R . Then #( ψ )Φ( ϕ )#( ψ ) − 1 ( µ ( x, σ, p )) = µ ( x, σ, η p + εt ) 16 BENO ˆ IT KLOECKNER Pr o of . — It fo llows from a dire c t co mputation: #( ψ )Φ( ϕ )#( ψ ) − 1 ( µ ( x, σ, p )) = #( ψ )Φ( ϕ )( µ ( εx − εv , σ, εp )) = #( ψ )( µ ( εx − εv , σ, η εp + t )) = µ ( x, σ, η p + εt ) W e are now a ble to prov e our main result. Pr o of of The o r em 1.1 . — First Le mma 5 .1 s ays that any iso metry o f W 2 ( R ) acts on ∆ 1 and ∆ 2 . Let L b e #(Iso m R ) and R be the subset o f Isom W 2 ( R ) consis ting o f isometries that fix ∆ 1 po in t wise. Then R is a nor mal subgroup of Isom W 2 ( R ). Let Ψ be an isometry of W 2 ( R ). It acts isometric a lly on ∆ 1 , thus there is an isometry ψ of R suc h that #( ψ )Ψ ∈ R . In par ticula r, Isom W 2 ( R ) = L R . Since L ∩ R is r educed to the iden tit y , we do have a se midir ect pro duct Isom W 2 ( R ) = L ⋉ R . According to P rop osition 4.2, each map Φ( ϕ ) : µ ( x, σ , p ) 7→ µ ( x, σ, ϕ ( p )) extends int o an isometry of C (∆ 2 ), which is W 2 ( R ) b y Pro pos ition 3.4. W e still denote by Φ( ϕ ) this extension. Prop osition 3 .4 also shows that an isometry of W 2 ( R ) is ent irely determined by its action on ∆ 2 . The description of R now fo llows from Lemma 5.2. If σ denotes the symmetry a round 0 ∈ R , then Φ( σ ) maps a measure µ ∈ W 2 ( R ) to its symmetric w ith r e spect to its cen ter of mass, th us pr eserves shap es. An y o ther ϕ ∈ Is o m( R ) is a transla tion o r the comp osition of σ and a tra nslation. By the exotic isometry flow of Isom W 2 ( R ) we mean the flow of isometries Φ t = Φ( ϕ t ) obtained when ϕ t : p → p + t is a translation. This flo w do es not preserve shap es as is seen from its expr ession in ∆ 2 . A t last, Lemma 5.3 gives the a sserted description o f the semidirect pro duct. 5.2. Beha viour of the exotic isom etry flo w. — The definition of Φ t is construc- tive, but not very explicit o utside ∆ 2 . On ∆ 2 , the flo w tends to put most of the mass on the right of the cen ter of mass, v ery close to it, and send a smaller and sma ller bit of mass far aw ay on the left. The flow Φ t preserves ∆ 3 as its elemen ts a r e the only ones to lie on a geo desic segment having bo th endp oints in ∆ 2 . Similarly , elemen ts of ∆ n are the only ones to lie on a g eo des ic segment having a n endpo in t in ∆ 2 and another in ∆ n − 1 , therefo re Φ t preserves ∆ n for all n . Direct computations ena ble one to find fo r m ulas for Φ t on ∆ n , but the express io ns one g e ts are no t so nice. F or example, if µ = 1 3 δ x 1 + 1 3 δ x 2 + 1 3 δ x 3 where x 1 6 x 2 6 x 3 , W ASSERSTEIN SP A CE OF EUCLIDEAN SP A CES 17 Cent er of ma ss Figure 7. Image of a point of ∆ 2 by Φ 2 (dashed) and Φ 3 (dotted). then Φ t ( µ ) = 1 1 + 2 t 2 δ x 1 + 1 3 (1 − t )( x 3 − x 1 )+ 1 3 (1 − t )( x 2 − x 1 ) + 3 2 t 2  1 + 1 2 t 2  (1 + 2 t 2 ) δ x 1 + 1 3 (1 − t )( x 3 − x 1 )+ 1 3 (1+ t − 1 + t )( x 2 − x 1 ) + 1 2 t 2 1 + 1 2 t 2 δ x 1 + 1 3 (1+2 t − 1 )( x 3 − x 1 )+ 1 3 (1 − t − 1 )( x 2 − x 1 ) In order to get so me in tuition ab out Φ t , let us pr ov e the following. Pr op osition 5. 4 . — L et µ b e any p oi nt of W 2 ( R ) and x its c enter of mass. If t go es to ±∞ , then Φ t ( µ ) c onver ges we akly to δ x . Pr o of . — W e shall o nly consider the cas e when t → + ∞ s ince the other o ne is sy m- metric. Let us star t with a lemma. L emma 5. 5 . — If γ t and ν t ar e in W 2 ( R ) and b o th c onver ge we akly to δ x when t go es to + ∞ , and if µ t is in the ge o desic se gment b etwe en γ t and ν t for al l t , then µ t c o nver ges we akly to δ x when t go es to + ∞ . Pr o of . — It is a direct consequence of the form of g eo desics: if γ t and ν t bo th charge an in terv a l [ x − η, x + η ] with a mass at leas t 1 − ε , then µ t m ust charge this in terv al with a mass a t least 1 − 2 ε . Now w e a re a ble to prov e the pr o po sition on lar ger and larger subsets of W 2 ( R ). First, it is ob vious on ∆ 2 . If it holds on ∆ n , the pr eceding Lemma together with Prop osition 3.3 implies that it holds on ∆ n +1 . T o prove it on the whole of W 2 ( R ), the density of the subset o f S n ∆ n consisting of meas ures having center of mas s x , and a diagonal pro cess are sufficient. 18 BENO ˆ IT KLOECKNER 6. Isometrie s: the highe r-di mensional case T o show that the e x otic isometry flow of W 2 ( R ) is exc eptional, let us consider the higher-dimensiona l case : ther e are is ometries o f W 2 ( R n ) that fix p oint wise the s et of Dirac masses, but there are not so many and they pres erve shap es. 6.1. Existence of non-trivial iso metries. — W e start with the existence of no n- trivial isometrie s , that ho wev er pr eserves shap es. Pr op osition 6. 1 . — If ϕ is a line ar isometry of R n , then the map Φ( ϕ ) : µ 7→ ϕ # ( µ − g ) + g wher e g denotes b oth the c enter of mass of µ and the c orr esp onding tr anslation, is an isometry of W 2 ( R n ) (se e figur e 8). Figure 8. Example of a non - trivial isometry that preserv es shap es. Note that we need ϕ to be linear, thus we do not g et as many no n-trivial isometries as in W 2 ( R ). Moreov er, all iso metries constructed this wa y pres e rve shap es. Pr o of . — W e only need to c heck the case of a bsolutely contin uous measur es µ, ν since they form a dens e subs et of W 2 ( R n ). In that cas e, ther e is a unique optimal coupling that is deterministic, g iv en by a map T : R n → R n such that T # ( µ ) = ν . Denote by g and h the centers of mass o f µ and ν . Let us show that there is a go o d coupling betw een µ ′ = Φ( ϕ )( µ ) and ν ′ = Φ( ϕ )( ν ). Let T ′ be the map defined by T ′ ( ϕ ( x − g ) + g ) = ϕ ( T x − h ) + h W ASSERSTEIN SP A CE OF EUCLIDEAN SP A CES 19 By construction T ′ # ( µ ′ ) = ν ′ . Moreov er the cost of the coupling (Id × T ′ ) # µ ′ is A = Z R n | ϕ ( x − g ) + g − ϕ ( T x − h ) − h | 2 µ ( dx ) = Z R n  | T x − x | 2 + 2 | g − h | 2 + 2( x − T x ) · ( h − g ) + 2 ϕ ( x − T x ) · ( g − h ) +2 ϕ ( h − g ) · ( g − h )  µ ( dx ) = d 2 W ( µ, ν ) + 2 | g − h | 2 + 2( g − h ) · ( h − g ) + 2 ϕ ( g − h ) · ( g − h ) +2 ϕ ( h − g ) · ( g − h ) = d 2 W ( µ, ν ) This shows that µ ′ and ν ′ are at distance at most d W ( µ, ν ). Applying the same reasoning to Φ( ϕ ) − 1 , w e g e t that d W ( µ ′ , ν ′ ) = d W ( µ, ν ) and Φ( ϕ ) is an isometry . 6.2. Semidi rect pro duct decomp osition. — The Dirac ma sses are the only mea - sures such that any geo desic issued from them can b e extended for all times (given any other mea sure, the geo desic pointing to any Dirac mass canno t b e ex tended pas t it, see in Sectio n 2.3 the para graph o n dilations). As a cons e quence, an iso metry of W 2 ( R n ) must globally preserve the set of Dira c ma sses As in the case of the line, if w e let L = # Isom( R n ) and R b e the set o f isometries of W 2 ( R n ) that fix each Dirac mass, then Iso m W 2 ( R n ) = L ⋉ R . W e proved above that R co n tains a copy of O( n ), and is in particular non-triv ial. Moreov er, if one co njugates a Φ( ϕ ) by some #( ψ ), where ϕ ∈ O ( n ) and ψ ∈ Isom R n , one gets the map Φ( ~ ψ ϕ ~ ψ − 1 ) (it is sufficien t to chec k this o n some measur e µ in the ea sy cases when ψ is a translation or fixes the center of mass of µ ). The r efore, the actio n o f L on O ( n ) ⊂ R in the semidir ect pro duct is as ass erted in The o rem 1.2. T o deduce Theo rem 1.2, we are thus left w ith proving that an iso metry that fixe s po in t wise all Dirac mas ses mu st b e of the form Φ( ϕ ) for some ϕ ∈ O( n ). 6.3. Measures supp orted on subspaces. — The following lemma will b e used to prove tha t iso metries of W 2 ( R n ) must pr eserve the prop erty of b eing suppo rted on a prop er subspace . L emma 6. 2 . — L et µ, ν ∈ W 2 ( R n ) , denote by g , h their c enters of mass and let σ = d W ( µ, δ g ) and ρ = d W ( ν, δ h ) . The e quality (7) d 2 W ( µ, ν ) = d 2 ( g , h ) + σ 2 + ρ 2 holds if and only if ther e ar e two ortho gonal affine su bsp ac es L and M such t hat µ ∈ W 2 ( L ) and ν ∈ W 2 ( M ) . 20 BENO ˆ IT KLOECKNER Pr o of . — Let us first pr ov e tha t d 2 ( g , h ) + σ 2 + ρ 2 is the cost B of the indep endent coupling Π = µ ⊗ ν : B := Z R n × R n d 2 ( x, y )Π( dxdy ) = Z R n × R n | ( x − g ) − ( y − h ) + ( g − h ) | 2 Π( dxdy ) = σ 2 + ρ 2 + d 2 ( g , h ) − 2  Z R n ( x − g ) µ ( dx )  ·  Z R n ( y − h ) ν ( dy )  +2( g − h ) ·  Z R n ( x − g ) µ ( dx )  − 2( g − h ) ·  Z R n ( y − h ) ν ( dy )  = σ 2 + ρ 2 + d 2 ( g , h ) since by definition g = R xµ ( dx ) and h = R y µ ( dy ). As a consequence, (7) holds if and only if the indep endent coupling is o ptima l. If µ has tw o p oint x, y in its supp ort a nd ν ha s tw o points z , t in its supp ort such that ( xy ) is not orthogo nal to ( z t ), then either ( x − y ) · ( z − t ) < 0 or ( x − y ) · ( t − z ) < 0. Then by cyclica l monoto nicit y (see Lemma 2.1) the s uppor t of an optimal coupling cannot contain ( x, z ) and ( y , t ) (in the first cas e) o r ( z , x ) a nd ( y , t ) (in the seco nd case) and thus cannot b e the indep endent coupling. As a consequence, (7) holds if and only if µ and ν are supp orted on tw o o rthogonal affine subspaces. L emma 6. 3 . — Isometries of W 2 ( R n ) send hyp erplane su pp orte d-me a sur es on hy- p erplane supp orte d me asur es. Mor e over, if two me asur es ar e supp orte d on p ar al lel hyp erplanes, then their images by any isometry ar e su pp orte d by p ar al lel hyp erplanes Pr o of . — Let µ ∈ W 2 ( R n ) be supported by some h yper plane H a nd Φ be an iso metry of W 2 ( R n ). Let ν be any mea sure that is supp orted on a line o rthogonal to H , and that is not a Dirac mass. Then (7) holds (whith the same notation as ab ov e). Let µ ′ and ν ′ denote the imag es o f µ a nd ν by Φ. W e k no w that Φ must ma p δ g and δ h to Dira c masses δ g ′ and δ h ′ . Since the cen ter of mass of an element of W 2 ( R n ) is uniquely defined a s its pro jection on the set of Dirac ma sses, g ′ is the cen ter of mass of µ ′ and h ′ is that of ν ′ . W e ge t, denoting by σ ′ and ρ ′ the distances of µ ′ and ν ′ to their centers of mass, d 2 W ( µ ′ , ν ′ ) = d 2 W ( µ, ν ) = d 2 ( g , h ) + σ 2 + ρ 2 = d 2 ( g ′ , h ′ ) + σ ′ 2 + ρ ′ 2 which implies that µ ′ and ν ′ are suppor ted o n or thogonal subspaces L and M of R n . Since ν ′ is not a Dirac mass, M is not a p oint and L is co n tained in some h yper pla ne H ′ . Moreov er, if µ 1 is another measure supp orted on a hyperpla ne pa rallel to H , then its image µ ′ 1 is supp orted on so me subspace orthogo nal to M . It follows tha t we can find a hyperpla ne pa rallel to H ′ that contains the supp ort of µ ′ 1 . W ASSERSTEIN SP A CE OF EUCLIDEAN SP A CES 21 W e are know equipp ed to end the pr o of of Theo rem 1.2 by induction o n the di- mension. 6.4. Non-existe nce of exotic iso m etries: case of the plane. — Given a line L ⊂ R 2 , denote by W 2 ( L ) the subset of W 2 ( R 2 ) cons isting of all meas ures whose suppo rt is a subset of L . An optimal coupling betw een tw o p oints of W 2 ( L ) m ust hav e its supp ort in L × L , thus W 2 ( L ) endo wed with the r estriction o f the distance o f W 2 ( R 2 ) is isometric to W 2 ( R ). Mor e precisely , given an y isometry ψ : R → L , w e get an isometry ψ # : W 2 ( R ) → W 2 ( L ). Lemma 6.3 ensures that a n y isometry Φ maps a line-supp orted measure to a line- suppo rted meas ur e, and that the v a rious measur e s in W 2 ( L ) are mapp ed to measures suppo rted on par allel lines. W e a s sume from now on that Φ fixes each Dirac mass and, up to compos ing it with some Φ( ϕ ), that it preser ves globally W 2 ( L ) for s ome L (the axis R × { 0 } say). W e can moreover assume that its restriction to W 2 ( L ) is a Φ t for some t . L emma 6. 4 . — L et Φ b e an isometry of W 2 ( R 2 ) that fi xes Dir ac masses, pr eserves glob al ly W 2 ( L ) and such that its r estriction to t his subsp a c e is the time t of the exotic isometric flow. Then t must b e 0 and up to c o mp osi ng with some Φ( ϕ ) , we c an assume that Φ pr e serves W 2 ( M ) for al l line M . Pr o of . — W e ident ify a measure µ ∈ W 2 ( R ) with its ima ge b y the usual embedding that identifi es R with the a xis L . W e deno te b y θ L the r otate of L by an angle θ around the orig in. Denote by µ ( x, σ, p, θ ) the combination of tw o Dirac mas s es that is the image o f µ ( x, σ, p ) if θ = 0, and its rotate around x by an angle θ otherwise. If θ 6 π / 2, o ne gets (8) d 2 W ( µ (0 , 1 , p, 0) , µ (0 , 1 , q , θ )) = 2 − 2 e | p − q | cos θ This shows in particular that the mea sures suppo rted on θ L and with cen ter of mass 0 must b e mapp ed to mea sures supp orted on ± θL . Up to comp osing with Φ( ϕ ) where ϕ is the orthogo nal sy mmetry with resp ect to L , w e can assume that the measur es supp or ted on π 3 L and with center of mass 0 are ma pped to meas ures suppo rted on π 3 L . Then the mea sures supp orted on θ L and with center of mass 0 must b e mapp ed to meas ur es suppor ted on a line that cr osses L and π 3 L with angles ± θ and ± ( θ − π 3 ). They are therefor e mapped to measures supp orted on θ L . Using the sa me argument and Lemma 6.3, we get that Φ m ust preserve W 2 ( M ) for all lines M . Moreov er, from equation (8) w e deduce that if the res triction of Φ to W 2 ( L ) is the time t of the ex otic isometric flow, then for a ll θ < π / 2 its restriction to θ L also is. But applying the same rea soning to π 3 L and 2 π 3 L , then to 2 π 3 L and π L = L we see that the restriction of Φ to W 2 ( L ) with the r everse d orientation must b e the time t of the exotic iso metr ic flow. This implies t = − t thus t = 0. The case n = 2 of Theor em 1.2 is now r educed to the following. 22 BENO ˆ IT KLOECKNER L emma 6. 5 . — If an isometry Φ of W 2 ( R 2 ) fixes p ointwise the set of line-su pp orte d me asur es, then it mu st b e the identity. Pr o of . — This is a co nsequence o f Radon’s theorem, which asser ts that a function (compactly supp orted and smo oth, say) in R n is ch aracter ized b y its integrals alo ng all hyperplanes [ 14 ] (also see [ 8 ]). Given µ ∈ W 2 ( R 2 ), one can determine by purely metr ic means its or thogonal pro jection on any fixed line L : it is its metric pro jection, that is the unique ν supp orted on L that minimizes the distance d W ( µ, ν ). Now if µ has smo oth density and is compactly supp orted, then the in tegral of its density along any line L is ex actly the density at p oint M ∩ L o f its pro jectio n onto any line M orthogona l to L . Therefore, Φ must fix every measure µ ∈ W 2 ( R 2 ) that ha s a smo oth density a nd is compactly supp orted. They form a dense s et of W 2 ( R 2 ) th us Φ must be the identit y . 6.5. Non-existe nce of e xotic i s ometries: g e neral case. — W e end the pr o o f of Theo rem 1 .2 by an induction o n the dimension. Ther e is nothing new compar e d to the case o f the plane, so we stay sketch y . Let Φ b e an isometry of W 2 ( R n ) that fixes po in twise the Dir ac mas ses. It must map every hyp e r plane-supp orted measure to a h yper plane-supp orted measure. Using a non-tr ivial isometry , we can ass ume that for some h yp erplane L , Φ glo bally preserves the set W 2 ( L ) of measures supp orted o n L . Thanks to the induction hyp othesis, we can comp ose Φ with ano ther non-tr ivial iso metr y to ensure that Φ fixes W 2 ( L ) point wise. Let µ b e a measure supp orted on s ome hyper plane M 6 = L . Let M ′ be a h yp erplane suppo rting Φ( µ ). Then as in the cas e of the plane, it is easy to show that the dihedral angle o f ( L, M ) equals that of ( L, M ′ ). Mo reov er, all measures supp orted on L ∩ M are fix ed by Φ, and we co nclude that M ′ = M (up to co mp ositio n with Φ( ϕ ) wher e ϕ is the or thogonal symmetry with r e spect to L ). The same arg ument shows that Φ preserves W 2 ( M ) for all hyperplanes M . A measure supp orted o n M is deter mined, if its dihedral angle with L differ en t from π / 2 , by its or thogonal pro jection o n to L . since Φ fixes W 2 ( L ) p oint wise, it m ust fix W 2 ( M ) p oint wise as well. When M is orthogo nal to L , the use o f a thir d hyperplane not or thogonal to M nor L yields the same conclusio n. Now that we know tha t Φ fixes every hyper plane-supp orted measur e, we ca n use the Radon Theor em to conclude that it is the identit y . 7. Ranks One usually defines the r ank of a metric s pace X as the supremum of the set of po sitive integers k such that there is an iso metric embedding of R k int o X . As a consequence of P r op osition 3.6, we ge t the following r esult announced in the int ro duction. The or em 7. 1 . — The sp a c e W 2 ( R n ) has r ank n . W ASSERSTEIN SP A CE OF EUCLIDEAN SP A CES 23 Pr o of . — An isometric e mbedding e : R n +1 → W 2 ( R n ) must map a geo desic to a geo desic, since they a re precisely those curves γ satisfying d ( γ ( t ) , γ ( s )) = v | t − s | for some co nstant v . The union of co mplete geo desics through any po in t µ in the image o f e would contain a copy of R n +1 , but P rop osition 3.6 shows that this union is isometric to R n . How ev er, one ca n define less r estrictive notions o f rank as follows. Definition 7.2 . — Let X b e a Polish space. The semi-glob al r ank of X is defined as the supremum of the set of p ositive integers k such tha t for all r ∈ R + , there is an isometric embedding of the ball of r adius r of R k int o X . The lo ose r ank of X is defined as the supremum of the set of p ositive integers k such that there is a qua si-isometric embedding o f Z k int o X . Let us recall that a map f : Y → X is s aid to b e a quasi-is ometric embedding if there are cons tan ts C > 1 , D > 0 s uc h that for all y , z ∈ Y the following holds : C − 1 d ( y , z ) − D 6 d ( f ( y ) , f ( z )) 6 C d ( y , z ) + D . The notion o f lo ose rank is r elev a n t in a larg e class of metric spaces, including discrete spaces (the Go rdian spa ce [ 7 ], or the Cayley gr aph of a finitely pr esented group for ex ample). W e chose not to call it “coar se rank” due to the previous use of this term by Kap ovic h, K le ine r and Leeb. The semi-global rank is mo tiv ated by the fo llowing simple result. Pr op osition 7. 3 . — A ge o desic sp ac e X that has semi-glob al r ank at le ast 2 is not δ -hyp erb olic. Pr o of . — Since X contains euclidean disks o f a rbitrary radius, it also con tains eu- clidean e quilateral tria ngles of arbitra r y diameter . In such a triangle, the maximal distance betw een a point of a n edg e a nd the other edges is prop or tional to the diam- eter, thus is un bo unded in X . Pr op osition 7. 4 . — The semi-glob al r ank and the lo ose r ank of W 2 ( R ) ar e infinite. Pr o of . — Conside r the subset R k 6 = { ( x 1 , . . . , x k ); x 1 6 x 2 6 · · · 6 x k } of R k . It is a closed, conv ex cone. Moreov er the ma p R k 6 → W 2 ( R ) ( x 1 , . . . , x k ) 7→ X 1 k δ x i is an iso metric em b e dding. Since R k 6 contains arbitra rily lar ge balls, W 2 ( R ) must hav e infinite semi-glo bal rank. 24 BENO ˆ IT KLOECKNER Moreov er, since R k 6 is a conv ex cone of no n-empt y interior, it c o n tains a circular cone. Such a cir cular cone is conjuga te by a linear (and th us bi-Lipschitz) map to the cone C = { x 2 1 = X i > 2 x 2 i } . Now the vertical pr o jection fro m { x 1 = 0 } to C is bi-Lipschitz. There is therefore a bi-Lipschitz em b edding of R k − 1 in W 2 ( R ) and, a fortiori , a quasi-isometric embedding of Z k − 1 . Therefore W 2 ( R ) has infinite lo ose rank. 7.1. Ranks of other spaces. — The ra nks of W 2 ( R ) hav e an influence on those of many spa ces due to the following lemma. L emma 7. 5 . — If X and Y ar e Polish ge o desic sp a c es, any isometric emb e dding ϕ : X → Y induc es an isometric emb e d ding ϕ # : W 2 ( X ) → W 2 ( Y ) . As usual, ϕ # is defined by: ϕ # µ ( A ) = µ ( ϕ − 1 ( A )) for all mea surable A ⊂ Y . Pr o of . — Since ϕ is is ometric, for any µ ∈ W 2 ( X ), ϕ # µ is in W 2 ( Y ). Moreover any optimal transp o rtation pla n in X is mapp ed to an optimal transp ortatio n plan in Y (note that a coupling be tw een t wo mea s ures with supp ort in ϕ ( X ) must hav e its suppo rt contained in ϕ ( X ) × ϕ ( X )). In tegrating the equality d ( ϕ ( x ) , ϕ ( y )) = d ( x, y ) yields the desire d result. Cor ol lary 7.6 . — If X is a Polish ge o desic sp ac e that c ontains a c omplete ge o desic, then W 2 ( X ) has infi nite semi-glob al r ank and infi nite lo ose r ank. As a c onse qu enc e, W 2 ( X ) is not δ -hyp erb olic. Pr o of . — F ollows from the preceding Lemma, Prop osition 7.4 and Pro pos ition 7.3. This obviously a pplies to W 2 ( R n ). One could hop e that in Hadamar d spa ces, the pro jection to the center of mass P : W 2 ( X ) → X could give a higher b ound on the r ank o f W 2 ( X ) by means of that of X . How ev er, P need not map a geo desic on a geo desic. F or ex a mple, if one cons ide r on the real hyperb olic plane R H 2 the measures µ t = 1 / 2 δ p + 1 / 2 δ γ ( t ) where p is a fixed p oint and γ ( t ) is a geo desic, then µ t is a geo desic of W 2 ( R H 2 ) that is mapp ed by P to a c ur ve with the sa me endp oints than γ , but is different from it. Therefo r e, it cannot b e a geo desic. 8. Op en problem s Since the higher-dimensio nal Euclidean s pa ces ar e more rigid (has few non-trivia l isometries) than the line, we expec t o ther spaces to b e even mor e r igid. W ASSERSTEIN SP A CE OF EUCLIDEAN SP A CES 25 p γ ( t ) P ( µ t ) Figure 9. The pro jection P maps a geodesic of W 2 ( R H 2 ) to a non-geodesic curve (dashed) in R H 2 . Question 1 . — Do es it exists a Polish (or Hada mard) space X 6 = R such that W 2 ( X ) admits exotic is o metries? Do e s it exis ts a Polish (or Hadamar d) space X 6 = R n such that W 2 ( X ) admits non-trivial isometries ? 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[20] , Optimal tr ans p ort old and new , Grundlehren der Mathematisc hen Wis- sensc haften [F undamental Principles of Mathematical S ciences], vol . 338, Sp ringer- V erlag, Berlin, 2009. Beno ˆ ıt Kloeckner , Institut F ourier, 100 rue des Maths, BP 74, 38402 St Martin d’H` eres, F rance • E-mail : bkloeckn@four ier.ujf-grenoble.fr Url : http://www-fo urier.ujf -grenoble.fr/~bkloeckn/