A discrete version of the Brunn-Minkowski inequality and its stability

In the ﬁrst part of the paper, we deﬁne an approximated Brunn-Minkowski inequality which generalizes the classical one for metric measure spaces. Our new deﬁnition, based only on properties of the distance, allows also us to deal with discrete metric measure spaces. Then we show the stability of our new inequality under convergence of metric measure spaces. This result gives as corollary the stability of the classical Brunn-Minkowski inequality for geodesic spaces. The proof of this stability was done for related inequalities (curvature-dimension inequality, metric contraction property) but not for the Brunn-Minkowski one, as far as we know. Inthe second part of the paper, we show that every metric measure space satisfying the classical Brunn-Minkowski inequality can be approximated by discrete metric spaces with some approximated Brunn-Minkowski inequalities


Introduction
Let us recall some facts about the Brunn-Minkowski inequality.It was first established in R n for convex bodies by Brunn and Minkowski in 1887 (for more details about the inequality and its birth, one can refer to the great surveys [1,5] and the references therein).It can be read as follows: if K and L are convex bodies (compact convex sets with non-empty interior) of R n and 0 ≤ t ≤ 1 then where V n is the Lebesgue measure on R n and + the Minkowski sum which is given by for A and B two sets of R n .Equality holds if and only if K and L are equal up to translation and dilation.The Brunn-Minkowski inequality is a very powerful inequality with a lot of applications.For example it implies very quickly the isoperimetric inequality for convex bodies in R n which reads where K is a convex body of R n , B n the unit ball of R n and s n the surface area measure; with equality if and only if K is a ball.The Brunn-Minkowski inequality is valid not only for convex bodies but also for all non-empty compact sets and even for all non-empty Borel sets of R n .One way to prove it is to establish the Prekopa-Leindler functional inequality (see [1]) which applied to characteristic functions of sets gives the multiplicative Brunn-Minkowski inequality where K and L are two Borel sets of R n .By homogenity of the Lebesgue measure V n , it can be shown that this a priori weaker inequality is in fact equivalent to the n-dimensional one (1.1).The Brunn-Minkowski inequality has a very strong geometric content and it is natural to ask on which more general spaces than R n the inequality can be extended.
A first answer would be to change the measure: a log-concave measure on R n satisfies the multiplicative Brunn Minkowski inequality.
But to be able to get away from R n , we have to generalize the Minkowski sum.This can be done on geodesic spaces with the notion of s-intermediate sets.This notion is very useful in optimal transportation on geodesic spaces (see [3] for length and geodesic spaces and [11] for optimal transportation) and then appeared in some studies of optimal transportation on Riemmannian manifolds (see [4]).Following an idea of this last paper, for two subsets C 0 and C 1 of a metric space X and s ∈ [0, 1] we define the s-intermediate set between C 0 and C 1 by .
(1.4)This is the set spanned by all geodesics going from a point in C 0 to a point in C 1 .On R n , this set is exactly the Minkowski sum of (1 − s)C 0 and sC 1 .The authors in [4] use it only for a Riemannian manifold but it makes sense for all metric spaces even if it is interesting only for geodesic spaces.In this context we will say a metric measure space (X, d, m) satisfies the N -dimensionnal Brunn-Minkowski inequality if for all 0 ≤ s ≤ 1 and all C 0 , C 1 non-empty compact sets of X.We will refer in the sequel at (1.5) as the "classical" N -dimensionnal Brunn-Minkowski inequality.It is proven in [4] that a Riemannian manifold M of dimension n whose Ricci curvature is always non negative satisfies (1.5) with dimension N = n and with its canonical volume measure, i.e.

vol(C
for all non-empty compact sets C 0 and C 1 of M and with vol the canonical volume measure of the Riemannian manifold.Recently, there have been a lot of works on the geometry of metric measure spaces.Lott-Villani and Sturm have given independently a synthetic treatment of metric spaces having Ricci curvature bounded below (see [7,9,10]).All these works were motivated by the result of precompactness of Gromov: the class of Riemannian manifolds of dimension n and Ricci curvature bounded below by some constant k is precompact in the Gromov-Hausdorff metric.The notion of lower bound on Ricci curvature they develop for metric spaces generalizes the Riemannian one and is stable under Gromov-Hausdorff convergence.Their definition relies on convexity properties of relative entropy on the Wasserstein space of probability measures and is linked with optimal transportation.
Sturm ([10]) in this context defines a Brunn-Minkowski inequality with curvature bounded below by k.The meaning of this inequality may not be totally satisfactory.Indeed the inequality involves a parameter Θ which is whether the curvature is non-negative or negative.It corresponds to the minimal or maximal length of geodesics between the two compact sets C 0 and C 1 .But this Brunn-Minkowski inequality is a direct consequence of the dimension-curvature condition CD(k, N ) defined in this paper and it gives all the geometric consequences of the theory like a Bishop-Gromov theorem on the growth of balls.
There is another weak concept of lower bound of Ricci curvature which is known as metric contraction property (see [8,10,6]) and which is implied by this last Brunn-Minkowski inequality at least in the case of curvature bounded below by 0 and the m ⊗ m a.s.existence and uniqueness of geodesics between two points of the metric measure space (X, m, d).
As far as we know the stability of Brunn-Minkowski inequality (1.5) was not proven yet.This is the main result of our paper (corollary 2.4).
In section 2, we prove the result for the classical Brunn-Minkowski inequality (1.5) (which corresponds to the inequality defined by Sturm with curvature bounded below by 0) and then explain in a remark how we can extend our result to the general case with curvature bounded below by k.We introduce an approximated Brunn-Minkowski inequality which will be essential in the core of the proof of our main result and allows us to deal with discrete metric spaces.
In section 3, we show that every metric measure space satisfying a classical Brunn-Minkowski inequality can be approximated by discrete metric spaces with some approximated Brunn-Minkowski inequalities.
To avoid issues with sets of zero measure we will work only with metric spaces (X, d, m) where (X, d) is Polish and m a Borel measure on (X, d) with full support, i.e. that charges every open ball of X.

Stability of Brunn-Minkowski inequality
Definition 2.1.Given h ≥ 0 and N ∈ N, N ≥ 1, we say that a metric measure space (X, d, µ) satisfies the h-Brunn-Minkowski inequality of dimension N denoted by BM (N, h) if for each pair (C 0 , C 1 ) of non-empty compact subsets of X, we have: where C h s is defined by We call the set C h s the set of h(-approximated) s-intermediate points between C 0 and C 1 .The idea of introducing this set is due to [2].If X is a geodesic space and h = 0, this inequality is just the classical Brunn-Minkowski inequality.We shall often note BM (N ) instead of BM (N, 0).Remark that this definition is meaningful for discrete metric spaces.Observe also that if X satisfies BM (N, h) it also satisfies BM (N, h ) for all h ≥ h.In this work we use the following distance D between abstract metric measure spaces.We refer to [9] for its properties.
where d is a pseudo-metric on the disjoint union M M which coincides with d on M and with d on M and q a coupling of the measures m and m .
A pseudo-metric d on M M is a metric which coincides with d on M and with d on M but for which the property d(x, y) = 0 does not necessarily imply that x and y are equal.A coupling q of the measures m and m is a measure on M × M whose marginals are m and m .Theorem 2.3.Let (X n , d n , m n ) be a sequence of compact metric measure spaces which converges with respect to the distance D to another compact metric measure space (X, d, m).
Before proving this theorem, observe that a direct consequence is the stability of the classical Brunn-Minkowski inequality for compact geodesic spaces: Corollary 2.4.Let (X n , d n , m n ) be a sequence of compact geodesic spaces which converges with respect to the distance D to another compact metric measure space (X, d, m), then X is also a geodesic space.If (X n , d n , m n ) satisfies BM (N ) then (X, d, m) satisfies also BM (N ).
The fact that the limit set X is a geodesic space is well known (see [9]).We will prove theorem 2.3 only for compact sets of positive measure.The remarks following the proof will extend the inequality to all nonempty Borel sets.
The idea of the proof is quite simple.Take two non-empty compact subsets of the limit set X. Choose and fix a nearly optimal coupling of X n and X.Then one construct two compact sets of X n by dilating the first two sets of X in X n X with respect to the pseudo-distance dn of the coupling and taking the restriction to X n .
The fact which makes things work is that the sets built in X n have nearly the same mass as the initial ones in X.Now we can define an approximated s-intermediate set in X n and apply the Brunn-Minkowski inequality in X n .By the same construction as before, we construct a set in the limit set X from the s-intermediate set in X n without losing too much measure.To conclude we have to study the link between this last set and the set of approximate s-intermediate points between initial compact subsets of X.
Proof of Theorem 2.3.Let C 0 , C 1 be two compact sets of X of positive measure and let s ∈ [0, 1].Extracting a subsequence, we may assume that D(X n , X) ≤ 1 2n .By definition of D, there exists dn a pseudo-metric on X n X and q n a coupling of m n and m such that Xn×X d2 n (x, y)dq n (x, y) Let us first prove that they are non empty for n large enough.Indeed, for i = 0, 1, On the other hand, we have: Therefore, for i = 0, 1, So the sets A n,i are non empty for n large enough.Now since X n satisfies BM (N, h n ), introduce the set Ân,s ⊂ X n defined as in definition 2.1 by 0 and A n,1 .By definition of BM (N, h n ), we have

This is the set of all the h n s-intermediate points between A n,
We can now define C n,s ⊂ X by for all x ≥ 0, combining the inequalities (2.2), (2.3) and (2.4) give us: Now let us prove that the set C n,s of X is included in the set C hn+4εn s of all the h n + 4ε n s-intermediate points between C 0 and C 1 defined as in defintion 2.1 by Indeed, let y ∈ C n,s , by definition of this set, there exists x ∈ Ân,s so that dn (x, y) ≤ ε n .By definition of Ân,s , it follows that there exists (x n,0 , x n,1 ) ∈ A n,0 × A n,1 such that By definition of A n,i for i = 0, 1, there exists (y 0 , y 1 ) ∈ C 0 × C 1 with dn (x n,0 , y 0 ) ≤ ε n and dn (x n,1 , y 1 ) ≤ ε n .It follows: The sequence (h n + 4ε n ) n is converging to h.Extracting a subsequence, we may assume this sequence is monotone.There are two cases.The first one is when the extracting subsequence is non-decreasing.Then we have Letting n go to infinity gives the conclusion.
The second one, more interesting, is when the extracted subsequence is non-increasing.Then we have By compactness of C 0 and C 1 we can extract another subsequence so that y n,0 → y 0 ∈ C 0 and y n,1 → y 1 ∈ C 1 and we have The other inclusion is immediate.This intersection is non-increasing so which gives the conclusion Remark 2.5.BM (N ) is directly implied by the conditions CD(O, N ) of Sturm or Lott and Villani for all the compact sets with positive measure (in fact for all the Borel sets with positive measure) (see [10]).But if the measure m charges all the balls of the space then if the space satisfies BM (N ) for all its compact subsets with positive measure it satisfies also it for all its non-empty compact subsets.Indeed let (X, d, m) be a metric measure space where the measure m charges all the open balls.Assume X satifies BM (N ) for all its compact subsets with positive measure.Let C 0 , C 1 be non-empty compact subsets with m(C 0 ) = 0 and m(C 1 ) > 0 (the case m(C 0 ) = m(C 1 ) = 0 is trivial) and s ∈ [0, 1].Define H ε 0 = {y ∈ X; ∃x ∈ C 0 , d(x, y) ≤ ε}, then m(H ε 0 ) > 0. Define H ε s the set of all the s-intermediate points between H ε 0 and C 1 .By Brunn-Minkowski inequality we have: m(C 2ε s ) which gives the annonced result.Consequently, on a compact metric measure space where the measure charges all the open balls, the condition CD(0, N ) implies BM (N ) for all non-empty compact sets.
Remark 2.6.In Polish spaces, Borel measures are regular.It enables us to pass from compact sets to Borel ones.More precisely, if a Polish space satisfies BM (N, h) for all its compact subsets, it also satisfies it for all its Borel subsets.Therefore, if the spaces X n and X are only Polish (no more compact), the sets A n,i for i = 0, 1 defined as above may be no more compact.However they will be closed, therefore (2.3) will still stay true in this more general context.We can, consequently, drop the assumption of compactness of X n and X in theorem (2.3) and its corollary (2.4).
Remark 2.7.We can do the same for the Brunn-Minkowski inequality with curvature bounded below by k by using the definition given in [10].The only additional thing to do is to control the parameter Θ.But, with preceeding notations, we have Remark 2.8.We can prove also the same theorem for the multiplicative Brunn-Minkowski inequality (1.3).

Discretizations of metric spaces
Let (M, d, m) be a given Polish measure space.For h > 0, let M h = {x i , i ≥ 1} be a countable subset of M with M = i≥1 B h (x i ).Choose A i ⊂ B h (x i ) measurable and mutually disjoint such that i≥1 A i = M and x i ∈ A i .Such a construction always exist (see [2] and the references therein).Consider the measure m h on M h given by m h ({x i }) = m(A i ) for i ≥ 1.We call (M h , d, m h ) a h-discretization of (M, d, m).It is proved in [2] that if m(M ) < ∞ then

Definition 2 . 2 .
Let (M, d, m) and (M , d , m ) be two metric measure spaces, their D distance is given by

Remark 3 . 3 .
x j .Define C i ⊂ M , for i = 0, 1, by C i = {x ∈ M ; ∃x j ∈ H i , d(x j , x) ≤ h}.By the first point of the lemma, for i = 0, 1,m(C i ) ≥ m h (H i ).(3.3)Let C s ⊂ M be the set of all the s-intermediate points between C 0 and C 1 in the entire space M , i.e.C s = x ∈ M ; ∃(c 0 , c 1 ) ∈ C 0 × C 1 , d(x, c 0 ) = s d(c 0 , c 1 ) d(x, c 1 ) = (1 − s) d(c 0 , c 1 ).BM (N ) inequality on M gives usm 1/N (C s ) ≥ (1 − s)m 1/N (C 0 ) + s m 1/N (C 1 ).(3.4)As before by triangular inequality, we can see that C s is included in the set C3h s of 3h s-intermediate points between H 0 and H 1 in the whole space M .Therefore, the set H 4h s ⊂ M h of 4h s-intermediate points between H 0 and H 1 in the discrete space M h contains the restriction at M h of the h dilated of C s .By the second point of the lemma we havem h (H 4h s ) ≥ m(C s ).(3.5)Combining inequalities (3.3), (3.4) and (3.5) ends the proof of the theorem.The same proof shows that if (M, d, m) satisfies BM (N, k), then (M h , d, m h ) satisfies BM (N, k + 4h).
1/N (C 1 ).As before H ε s is included in C 2ε s the set of all 2ε s-intermediate points between C 0 and C 1 .As before ε>0 C 2ε s is an non-increasing intersection equal to C 0 s the set of all the exact s-intermediate points between C 0 and C