A discrete version and stability of Brunn Minkowski inequality
In the first part of the paper, we define an approximated Brunn-Minkowski inequality which generalizes the classical one for length spaces. Our new definition based only on distance properties allows us also to deal with discrete spaces. Then we show…
Authors: ** Michel Bonnefont **
A disrete v ersion and stabilit y of Brunn Mink o wski inequalit y Mi hel Bonnefon t Institut de mathématiques, Lab oratoire de Statistique et Probabilités, Univ ersité P aul Sabatier, 118 route de Narb onne, 31062 T oulouse, FRANCE No v em b er 16, 2018 Abstrat In the rst part of the pap er, w e dene an appro ximated Brunn- Mink o wski inequalit y whi h generalizes the lassial one for length spaes. Our new denition based only on distane prop erties allo ws us also to deal with disrete spaes. Then w e sho w the stabilit y of our new inequalit y under a on v ergene of metri measure spaes. This result giv es as a orol- lary the stabilit y of the lassial Brunn-Mink o wski inequalit y for geo desi spaes. The pro of of this stabilit y w as done for dieren t inequalities (ur- v ature dimension inequalit y , metri on tration prop ert y) but as far as w e kno w not for the Brunn-Mink o wski one. In the seond part of the pap er, w e sho w that ev ery metri measure spae satisfying lassial Brunn-Mink o wski inequalit y an b e appro ximated b y disrete spaes with some appro ximated Brunn-Mink o wski inequalities. 1 In tro dution Let us reall some fats ab out the Brunn-Mink o wski inequalit y . First the in- equalit y w as set in R n for on v ex b o dies b y Brunn and Mink o wski in 1887 (for more details ab out the inequalit y and its birth, one an refer to the great sur- v eys [1 , 5℄ and the referene therein). It an b e read as if K and L are on v ex b o dies (ompat on v ex sets with non empt y in terior) of R n and 0 < t < 1 then V n ((1 − t ) K + tL ) 1 /n ≥ (1 − t ) V n ( K ) 1 /n + tV n ( L ) 1 /n (1) where V n is the Leb esgue measure on R n and + the Mink o wski sum whi h is giv en b y A + B = { a + b , a ∈ A, b ∈ B } 1 for A and B t w o sets of R n . Equalit y holds if and only if K and L are equals up to translation and dilatation. Brunn-Mink o wski inequalit y is a v ery p o w erful inequalit y with a lot of ap- pliations. F or example it implies v ery qui kly the isop erimetri inequalit y for on v ex b o dies in R n whi h reads V n ( K ) V n ( B ) 1 /n ≤ s ( K ) s ( B ) 1 / ( n − 1) (2) where K is a on v ex b o dy of R n and s the surfai measure, with equalit y if and only if K is a ball. The Brunn-Mink o wski inequalit y is not only true for on v ex b o dies but also for all ompat sets and ev en for all measurable sets of R n (with the little diult y that the Mink o wski sum of t w o mesurable sets is not neessary mea- surable). One w a y to pro v e it is to pro v e a funtional inequalit y kno wn as Prek opa-Leindler inequalit y whi h applied to harateristi funtions of sets giv es the m ultipliativ e Brunn-Mink o wski inequalit y V n ((1 − t ) K + tL ) ≥ V n ( K ) 1 − t V n ( L ) t (3) where V n is the Leb esgue measure on R n , K and L t w o measurable sets of R n . By homogenit y of the v olume V n , it an b e sho wn that this a priori w eak inequalit y is in fat equiv alen t to the n -dimensional one (1). All this w as to sho w that Brunn-Mink o wski inequalit y has a v ery geometri meaning and it is natural to ask on whi h more general spaes than R n the inequalit y an b e extended. One rst answ er is w e an hange the measure, for example a measure log- ona v e on R n satisfy m ultipliativ e Brunn Mink o wski. But to b e able to quit R n , w e ha v e to generalize the Mink o wski sum. This an b e done on length spaes b y using ideas of optimal transp ortation (refer to [3 ℄ for length spae, [7 ℄ for optimal transp oration, and for exemple [4℄ for this generalisation). F ollo wing an idea of this pap er, for t w o sets K and L of a metri spae X w e dene what w e are going to all the s -in termediate set b et w een K and L b y Z s ( K, L ) = z ∈ X ; ∃ ( k , l ) ∈ K × L , d ( k , z ) = sd ( k , l ) d ( z , l ) = (1 − s ) d ( k , l ) (4) This set will pla y the role set of baryen ters of the Mink o wski sum. In fat the authors in [4 ℄ use it only for a Riemannian manifold but it mak es sense for all metri spaes ev en if it is in teresting only for length spae. In this on text w e will sa y a metri measure spae ( X, d, m ) satises the N -dimensionnal Brunn- Mink o wski inequalit y if m 1 / N ( Z s ( K, L )) ≥ (1 − s ) m 1 / N ( K ) + s m 1 / N ( L ) (5) for all 0 < s < 1 and K , L ompats of X . W e will refer in the sequel at (5) as the "lassial" N -dimensionnal Brunn-Mink o wski inequalit y . It is pro v en in 2 [4 ℄ that a Riemannian manifold M of dimension n whose Rii's urv ature is alw a ys non negativ e satises (5) with dimension N = n and with the anonial v olume of the Riemannian manifold as measure, i.e. v ol ( Z s ( K, L )) 1 /n ≥ (1 − s ) v ol ( K ) 1 /n + s vol ( L ) 1 /n (6) for all ompats K , L of M where v ol denotes the anonial v olume of the Riemannian manifold. In fat they obtain more preise results on funtionnal inequalities lik e Prek opa-Leindler and Borell-Brasamp-Lieb inequalities. Reen tly , there ha v e b een a lot of w orks on geometry of metri measure spaes. Lott-Villani and Sturm ha v e giv en indep enden tly a syn theti treatmen t of metri spaes ha ving Rii urv ature b ounded b elo w b y k (see [7, 9 , 10 ℄). All these w orks b egan b y the result of preompatness of Gromo v: the lass of Riemannian manifolds of dimension n and Rii urv ature b ounded b elo w b y some onstan t k is preompat for a Gromo v-Hausdor metri. So the notion they dev elop for metri spaes has to generalize the one for Riemannian mani- folds and has to b e stable b y Gromo v-Hausdor on v ergene. Their denition is ab out on v exit y prop erties of relativ e en trop y on the W asserstein spae of probabilit y and is link ed with optimal transp ortation. Sturm in this on text denes a Brunn Mink o wski inequalit y with urv ature k (see [10 ℄). The meaning of this inequalit y ma y b e not totally satisfatory . Indeed the inequalit y is dep ending on parameter Θ whi h equals inf k ∈ K,l ∈ L d ( k , l ) or sup k ∈ K,l ∈ L d ( k , l ) whether the urv ature is p ositiv e (or n ull) or negativ e. It orresp onds to the minimal or maximal length of geo desis b et w een the t w o ompats K and L . Ho w ev er this is a diret impliation from its dimension- urv ature ondition C D ( k , N ) and this is this inequalit y that giv es all the geo- metri onsequenes of their theory lik e for example a Bishop-Gromo v theorem on the gro wth of balls. There is another w eak onept of urv ature whi h is kno wn as metri on- tration prop ert y (see [8 , 10 , 6℄) and whi h is implied b y this Brunn-Mink o wski inequalit y at least in the ase of urv ature 0 and the m ⊗ m a.s. uniqueness of geo desis b et w een t w o p oin ts of X . As far as I kno w stabilit y of Brunn-Mink o wski inequalit y w as not pro v en y et. This is the most in teresting result w e ha v e in the pap er (orollary 2.4 ). F or simpliit y w e will w ork only with the lassial Brunn-Mink o wski (i.e. with urv ature 0) and explains ho w to extend our results in the general ase, with urv ature k , in a remark. F or doing this w e in tro due an appro ximated Brunn mink o wski inequalit y sine w e need it during the pro of. This fat is in teresting in itself sine it allo ws us to deal with disrete spaes. In the seond part of the pap er w e sho w that ev ery metri measure spae satisfying lassial Brunn-Mink o wski inequalit y an b e appro ximated b y disrete spaes with some appro ximated Brunn-Mink o wski inequalities. T o a v oid some problems b et w een sets with zero measure w e will w ork only with metri spaes ( X, d, m ) where ( X, d ) is P olish and m a Borel measure on ( X, d ) with full supp ort, i.e. that harges ev ery ball of X . 3 2 Stabilit y of Brunn-Mink o wski inequalit y Denition 2.1. Given h ≥ 0 and N ∈ N , N ≥ 1 , we say that a metri me a- sur e sp a e ( X, d, µ ) satises the h Brunn-Minkowski ine quality of dimension N denote d by B M ( N , h ) if ∀ C 0 , C 1 ⊂ X omp ats, ∀ s ∈ [0 , 1] , we have: µ 1 / N ( C s h ) ≥ (1 − s ) µ 1 / N ( C 0 ) + s µ 1 / N ( C 1 ) (7) wher e C h s = x ∈ X/ ∃ ( x 0 , x 1 ) ∈ C 0 × C 1 / | d ( x 0 , x ) − sd ( x 0 , x 1 ) | ≤ h | d ( x, x 1 ) − (1 − s ) d ( x 0 , x 1 ) | ≤ h (8) W e all the set C h s the set of h (-appro ximated) s -in termediate p oin ts b et w een C 0 and C 1 . One an note that if X is a geo desi spae and h = 0 , it giv es ba k the lassial Brunn-Mink o wski inequalit y for geo desi spaes. W e shall often note B M ( N ) instead of B M ( N , 0 ) . Another remark to b e done is that this denition an b e used for disrete spaes. One an also note that if X satisfy B M ( N , h ) it will also satisfy B M ( N , h ′ ) for all h ′ ≥ h . In these notes w e use the follo wing distane D b et w een abstrat metri measure spaes. W e refer to [9 ℄ for its prop erties. Denition 2.2. L et ( M , d, m ) and ( M ′ , d ′ , m ′ ) b e two metri me asur e sp a es, their distan e D is given by D (( M , d, m ) , ( M ′ , d ′ , m ′ )) = inf ˆ d,q Z M × M ′ ˆ d 2 ( x, x ′ ) dq ( x, y ) 1 / 2 wher e ˆ d is a pseudo metri on M ⊔ M ′ whih oinides with d on M and with d ′ on M ′ and q a oupling of the me asur es m and m ′ . Theorem 2.3. L et ( X n , d n , m n ) b e a se quen e of omp at metri me asur e sp a es whih onver ges with r esp e t to the distan e D to another omp at met- ri me asur e sp a e ( X, d, m ) . If ( X n , d n , m n ) satises B M ( N , h n ) with h n → h when n go es to innity, then ( X, d, m ) satises B M ( N , h ) . In partiular for ompat geo desi spaes it implies diretly the stabilit y of the lassial Brunn-Mink o wski inequalit y with resp et to the D -on v ergene: Corollary 2.4. L et ( X n , d n , m n ) b e a se quen e of omp at ge o desi sp a es whih onver ges with r esp e t to the distan e D to another omp at metri me a- sur e sp a e ( X, d, m ) , then X is also a ge o desi sp a e. If ( X n , d n , m n ) satises B M ( N ) then ( X, d, m ) satises also B M ( N ) . W e will mak e the pro of of theorem 2.3 only for ompat sets of stritly p ositiv e measure. The remarks after the pro of will giv e the inequalit y for all 4 mesurable sets. The idea of the pro of is quite simple. W e ho ose t w o ompats of the limit set X . Then w e ho ose a go o d oupling of X n and X and w e onstrut t w o om- pats of X n b y dilating these ompats with resp et to the pseudo-distane of the oupling and taking the restrition of this t w o sets with X n . The fat whi h mak es things w ork is that the op eration w e did do esn't lose t w o m u h measure. So, w e an dene a s -in termediate set in X n and apply Brunn-Mink o wski in- equalit y in X n . By the same onstrution as b efore, w e onstrut a set in the limit set X from the s -in termediate set in X n without lo osing a lot of measure. T o onlude w e ha v e to study the link b et w een this set and set of appro ximate s -in termediate p oin ts b et w een initial ompats. Pro of of Theorem 2.3 Let C 0 , C 1 t w o ompats of X of stritly p ositiv e measure. Let s ∈ [0 , 1] . Cho ose n so that D ( X n , X ) ≤ 1 2 n . By denition of D , there exists ˆ d a pseudo-metri on X n ⊔ X and q a oupling of m n and m so that Z X n × X ˆ d 2 ( x, y ) dq ( x, y ) 1 / 2 ≤ δ n = 1 n F or ε n > 0 dene C ε n n,i = { x ∈ X n / ˆ d ( x, C i ) ≤ ε n } for i = 1 , 2 , these are ompats of X n . They are indeed not empt y for n large enough and ε n w ell hosen, sine b eing of stritly p ositiv e measure as w e will see it. W e ha v e m ( C 0 ) = q ( X n × C 0 ) = q ( C ε n n, 0 × C 0 ) + q ( { X n \ C ε n n, 0 } × C 0 ) But if ( x, y ) ∈ { X n \ C ε n n, 0 } × C 0 , then ˆ d ( x, y ) ≥ ε n , so q ( { X n \ C ε n n, 0 } × C 0 ) ≤ Z { X n \ C ε n n, 0 }× C 0 ˆ d 2 ( x, y ) ε n 2 dq ( x, y ) ≤ δ 2 n ε n 2 whi h equals 1 n for δ n = 1 n and ε n = 1 √ n . On the other hand, w e ha v e: m n ( C ε n n, 0 ) = q ( C ε n n, 0 × X ) ≥ q ( C ε n n, 0 × C 0 ) Consequen tly , m n ( C 1 √ n n, 0 ) ≥ m ( C 0 ) − 1 n (9) and iden tially m n ( C 1 √ n n, 1 ) ≥ m ( C 1 ) − 1 n . (10) 5 No w onsider the set C ε n ,h n n,s ⊂ X n dened as in the denition (2.1 ) b y C ε n ,h n n,s = x ∈ X n / ∃ ( x n, 0 , x n, 1 ) ∈ C ε n n, 0 × C ε n n, 1 / | d ( x n, 0 , x ) − sd ( x n, 0 , x n, 1 ) | ≤ h n | d ( x, x n, 1 ) − (1 − s ) d ( x n, 0 , x n, 1 ) | ≤ h n This is the set of all the h n s -in termediate p oin ts b et w een C ε n n, 0 and C ε n n, 1 . Sine X n satises B M ( N , h n ) , m 1 n n ( C ε n ,h n n,s ) ≥ (1 − s ) m 1 / N n ( C ε n n, 0 ) + s m 1 / N n ( C ε n n, 1 ) (11) W e an no w dene C ε n ,h n s ⊂ X b y C ε n ,h n s = { y ∈ X , ∃ x ∈ C ε n ,h n n,s ˆ d ( x, y ) ≤ ε n } Similary to (9) w e ha v e m ( C 1 √ n ,h n s ) ≥ m n ( C ε n n,s ) − 1 n (12) No w sine ( x − 1 n ) 1 / N + ≥ x 1 / N − ( 1 n ) 1 / N for all x ≥ 0 , om bining the inequalities (9), (10 ), (12 ) and (11) giv e us, for ε n = 1 √ n , m 1 / N ( C ε n ,h n s ) ≥ m 1 / N n ( C ε n ,h n n,s ) − ( 1 n ) 1 / N ≥ (1 − s ) m 1 / N n ( C ε n n, 0 ) + s m 1 / N n ( C ε n n, 1 ) − ( 1 n ) 1 / N ≥ (1 − s ) m 1 / N ( C 0 ) + s m 1 / N ( C 1 ) − 2( 1 n ) 1 / N C ε n ,h n s is inluded in the set K h n +4 ε n s of all the h n + 4 ε n s -in termediate p oin ts b et w een C 0 and C 1 . Indeed, let y ∈ C ε n ,h n s , b y denition of this set, there exists x ∈ C ε n ,h n n,s so that ˆ d ( x, y ) ≤ ε n . By denition of C ε n ,h n n,s , it follo ws that there exists ( x n, 0 , x n, 1 ) ∈ C ε n n, 0 × C ε n n, 1 satisfying | d n ( x, x n, 0 ) − s d n ( x n, 0 , x n, 1 ) | ≤ h n | d n ( x, x n, 1 ) − (1 − s ) d n ( x n, 0 , x n, 1 ) | ≤ h n . There exists, b y denition of C ε n n,i for i = 1 , 2 , ( y 0 , y 1 ) ∈ C 0 × C 1 with ˆ d ( x n, 0 , y 0 ) ≤ ε n and ˆ d ( x n, 1 , y 1 ) ≤ ε n . It follo ws: | ˆ d ( y , y 0 ) − s ˆ d ( y 0 , y 1 ) | ≤ | ˆ d ( y , y 0 ) − ˆ d ( x, x n, 0 ) | + | ˆ d ( x, x n, 0 ) − s ˆ d ( x n, 0 , x n, 1 ) | + s | ˆ d ( y 0 , y 1 ) − ˆ d ( x n, 0 , x n, 1 ) | ≤ h n + 4 ε n . and | ˆ d ( y , y 1 ) − (1 − s ) ˆ d ( y 0 , y 1 ) | ≤ h n + 4 ε n . 6 The sequene ( h n + ε n ) n is on v erging to h . W e an extrat a monotone sequene from it whi h will still b e denoted b y h n + ε n . There are t w o ases. The rst one is when the extrating subsequene is non-dereasing. Then w e ha v e K h n +4 ε n s ⊂ K h s . So, for all n , m 1 / N ( K h s ) ≥ m 1 / N ( K h n +4 ε n s ) ≥ (1 − s ) m 1 / N ( C 0 ) + s m 1 / N ( C 1 ) − 2( 1 n ) 1 / N . Letting n go es to innit y giv es the onlusion. The seond one, more in teresting, is when the extrated subsequene is non- inreasing. Then w e ha v e K h s = \ n K h n +4 ε n s . Indeed if y ∈ T n K h n +4 ε n s , for all n ∈ N , ∃ ( y n, 0 , y n, 1 ) ∈ C 0 × C 1 so that | d ( y , y n, 0 ) − s d ( y n, 0 , y n, 1 ) | ≤ h n + 4 ε n | d ( y , y n, 1 ) − (1 − s ) d ( y n, 0 , y n, 1 ) | ≤ h n + 4 ε n . By ompatness of C 0 and C 1 w e an extrat another subsequene so that y n, 0 → y 0 ∈ C 0 and y n, 1 → y 1 ∈ C 1 and w e ha v e | d ( y , y 0 ) − s d ( y 0 , y 1 ) | ≤ h | d ( y , y 1 ) − (1 − s ) d ( y 0 , y 1 ) | ≤ h . The other inlusion is immediate. This in tersetion is non-inreasing so m 1 / N ( K h s ) = lim n →∞ m 1 / N ( K h n +4 ε n s ) whi h giv es the onlusion m 1 / N ( K h s ) ≥ (1 − s ) m 1 / N ( C 0 ) + s m 1 / N ( C 1 ) . Remark 1. B M ( N ) is diretly implied b y the ondition C D ( O , N ) of Sturm or Lott and Villani for the ompat sets with a stritly p ositiv e measure (in fat for mesurable sets with stritly p ositiv e measure) (see [10 ℄). But if the mea- sure m is harging all the balls of the spae and (if the spae is geo desi), then the fat of ha ving B M ( N ) for all the ompats subspae with stritly p ositiv e measure implies B M ( N ) for all ompat subspaes. Indeed if ( X, d, m ) saties B M ( N ) for all the ompat sets with a stritly p ositiv e measure and if the measure m is harging all the balls, if C 0 , C 1 are om- pats with m ( C 0 ) = 0 and m ( C 1 ) > 0 (the ase m ( C 0 ) = m ( C 1 ) = 0 is trivial) and s ∈ [0 , 1] . Dene C ε 0 = { y ∈ X, ∃ x ∈ C 0 /d ( x, y ) ≤ ε } , m ( C ε 0 ) > 0 . Dene H ε s the set of all the s -in termediate p oin ts b et w een C ε 0 and C 1 , By Brunn-Mink o wski inequalit y w e ha v e: m 1 / N ( H ε s ) ≥ (1 − s ) m 1 / N ( C ε 0 ) + s m 1 / N ( C 1 ) ≥ s m 1 / N ( C 1 ) 7 H ε s is inluded in K 2 ε s the set of all 2 ε s -in termediate p oin ts b et w een C 0 and C 1 . As b efore T ε> 0 K 2 ε s is an non-inreasing in tersetion equal to K 0 s the set of all the exat s -in termediate p oin ts b et w een C 0 and C 1 . So m ( K 0 s ) = lim ε → 0 K 2 ε s whi h giv es the annoned result. Consequen tly , on a metri measure spae where the measure harges all the balls, C D (0 , N ) implies B M ( N ) for all ompats whi h in turns implies M C P (0 , N ) 2. In P olish spaes, Borel measures are regular whi h p ermits to pass from ompat sets to measurable ones. More preisely , if a P olish spae satisfy B M ( N , h ) for all his ompat subsets, it also satises it for all his measur- able subsets. Therefore, if the spaes X n and X are only P olish (no more ompats), the sets C ε n n,i for i = 1 , 2 dened as ab o v e ma y b e no more ompats. Ho w ev er they will still b e measurable sine losed, so (11 ) will still sta y true in this more general on text. W e an, onsequen tly , drop the assumption of ompatness of X n and X in the theorem ( 2.3 ) and its orollarry (2.4 ). 3. W e an do the same for the Brunn-Mink o wski inequalit y with urv ature k b y using the denition giv en in [10 ℄. The only additional thing to do is to on trol the parameter Θ . But, with preeeding notations, w e ha v e | Θ( C 0 , C 1 ) − Θ( C ε n n, 0 , C ε n n, 1 ) | ≤ 2 ε n . 4. W e an pro v e also the same theorem for the m ultipliativ e Brunn-Mink o wski inequalit y (3). 3 Disretizations of metri spaes Let ( M , d, m ) b e a giv en P olish measure spae. F or h > 0 , let M h = { x i , i ≥ 1 } b e a oun table subspae of M with M = S i ≥ 1 B h ( x i ) . Cho ose A i ⊂ B h ( x i ) , x i ∈ A i m utually disjoin t and mesurable so that S i ≥ 1 A i = M . Con- sider the measure m h on M h giv en b y m h ( { x i } ) = m ( A i ) for i ≥ 1 . W e all ( M h , d, m h ) a disretization of ( M , d, m ) . It is pro v ed in [2 ℄ that if m ( M ) < ∞ then ( M h , d, m h ) D − → ( M , d, m ) . Theorem 3.1. If ( M , d, m ) satises B M ( N ) then ( M h , d, m h ) satises B M ( N , 4 h ) . The pro of is based on the t w o follo wing fats. Lemma 3.2. 1. If H ⊂ M h then m ( H h ) ≥ m h ( H ) (13) wher e H h = { x ∈ M , d ( x, H ) ≤ h } . 8 2. If A ⊂ M mesur able and A h = { x i ∈ M h , d ( x i , A ) ≤ h } then m h ( A h ) ≥ m ( A ) . (14) Pro of of lemma 3.2 First, let H ⊂ M h , w e ha v e m h ( H ) = X i/x i ∈ H m ( A i ) = m ( ⊔ i/x i ∈ H A i ) ≤ m ( H h ) sine ⊔ i/x i ∈ H A i ⊂ H h = { x ∈ M , d ( x, H ) ≤ h } . F or the seond p oin t, let A ⊂ M mesurable, dene A h as ab o v e, then m h ( A h ) = X i/x i ∈ A h m ( A i ) = m ( ⊔ i/x i ∈ A h A i ) ≥ m ( A ) sine ⊔ i/x i ∈ A h A i ⊃ A . Indeed if for some j , A j ∩ A 6 = ∅ then there exists a ∈ A with d ( x j , a ) ≤ h so x j ∈ A h . Pro of ot theorem 3.1 Let H 0 , H 1 b e t w o ompats of M h and s ∈ [0 , 1] . H 0 and H 1 onsist of a nite or oun table n um b er of p oin ts x j . Dene H h 0 , H h 1 ⊂ M b y H h i = { x ∈ M , ∃ x j ∈ H i /d ( x j , x ) ≤ h } for i = 1 , 2 . By the rst p oin t of the lemma, for i = 1 , 2 m ( H h i ) ≥ m h ( H i ) . (15) Let ( H h ) s ⊂ M b e the set of all the s -in termediate p oin ts b et w een H h 0 and H h 1 in the en tire spae M , i.e. ( H h ) s = x ∈ M , ∃ ( x 0 , x 1 ) ∈ H h 0 × H h 1 / d ( x, x 0 ) = s d ( x 0 , x 1 ) d ( x, x 1 ) = (1 − s ) d ( x 0 , x 1 ) B M ( N ) inequalit y on M giv es us m 1 / N (( H h ) s ) ≥ (1 − s ) m 1 / N ( H h 0 ) + s m 1 / N ( H h 1 ) . (16) As b efore b y triangular inequalit y , w e an see ( H h ) s is inlude in the set ˜ C 3 h s of 3 h s -in termediaire p oin ts in the whole spae M b et w een H 0 and H 1 . So the set ˜ H 4 h s ⊂ M h of 4 h s -in termediate p oin ts b et w een H 0 and H 1 in the disrete spae M h on tains the restrition at M h of the h dilated of ( H h ) s . By the seond p oin t of the lemma w e ha v e m h ( ˜ H 4 h s ) ≥ m (( H h ) s ) . (17) 9 Com bining inequalities (15), (16 ) and (17) ends the pro of of the theorem. Remark If ( M , d, m ) satises B M ( N , k ) then ( M h , d, m h ) satises B M ( N , k + 4 h ) . Referenes [1℄ F. Bar the , A utour de l'iné galité de Brunn-Minkowski . Ann. F a. Si. T oulouse Math. (6), ( 2003 ) v ol 12, 27178 [2℄ A.I. Bonioa t and K.T. Sturm , Mass tr ansp ortation and r ough urvatur e b ounds for disr ete sp a es . Preprin t [3℄ D. Bura go , Y. Bura go and S. Iv ano v , A ourse in metri ge ometry . Graduate Studies in Mathematis 33. 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Graduate Studies in Mathematis 58. Amer- ian Mathematial So iet y ( 2003 ) 10
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