On the infimum convolution inequality

Abstract

Abstract. We study the infimum convolution inequalities. Such an inequality was first introduced by B. Maurey to give the optimal concentration of measure behaviour for the product exponential measure. We show how IC inequalities are tied to concentration and study the optimal cost functions for an arbitrary probability measure µ. In particular, we prove an optimal IC inequality for product log-concave measures and for uniform measures on the n p balls. Such an optimal inequality implies, for a given measure, the central limit theorem of Klartag and the tail estimates of Paouris.

On the convex infimum convolution inequality with optimal cost function

“…If X is symmetric and the pair (X, ϕ) satisfies the ICI, then ϕ(x) ≤ Λ * X (x) for every x ∈ R n (see Remark 2.12 in [10]). In other words, Λ * X is the optimal cost function ϕ for which the ICI can hold.…”

“…Since this conclusion is obtained by testing (1.1) with linear functions, the same holds for the convex ICI. Following [10] we shall say that X satisfies (convex) IC(β) if the pair (X, Λ * X (·/β)) satisfies the (convex) ICI.…”

“…. , X n ) satisfies (convex) IC(max β i ) (see [11] and [10]). Therefore we have the following corollary.…”

Poincaré inequality in mean value for Gaussian polytopes

“…This inequality is an analog for convex bodies of the tensorisation property of Poincaré inequalities. Up to now, it has been verified for the unit-balls of l n p with p ∈ [1, ∞] [32,44], the regular simplex [5], convex bodies of revolution [23] and for volume perturbations thereof [39]; and it has almost been verified for unconditional convex bodies [30]. Note that the Poincaré inequality (1) for the function |.| 2 with the estimate coming from (3), states that…”

Relative entropy of cone measures and L p centroid bodies

“…In view of a result of R. Latala and J. Wojtaszczyk [27], Theorem 2.2 has another consequence: The floating body of a symmetric convex body K corresponds to a level set of the Legendre transform of the logarithmic Laplace transform on K.…”