Rogers–Shephard inequality for log-concave functions

Abstract

In this paper we prove different functional inequalities extending the classical Rogers-Shephard inequalities for convex bodies. The original inequalities provide an optimal relation between the volume of a convex body and the volume of several symmetrizations of the body, such as, its difference body. We characterize the equality cases in all these inequalities. Our method is based on the extension of the notion of a convolution body of two convex sets to any pair of log-concave functions and the study of some geometrical properties of these new sets.

Forward and Reverse Entropy Power Inequalities in Convex Geometry

“…An integral lifting of the Rogers-Shephard inequality was developed by Colesanti [46] (see also [2,5]). For a real non-negative function f defined in R d , define the difference function ∆f of f ,…”

Rogers-Shephard type inequalities for sections

“…A reverse inequality of this was discovered by Rogers and Shephard in the 1950s, the so-called Rogers-Shephard inequality (see [25,Theorem 1] and [29,Section 10.1]). The Rogers-Shephard inequality reads: given any convex body K ⊂ R n , (1) vol n (K − K) ≤ 2n n vol n (K),…”

“…In recent years, both the Brunn-Minkowski inequality and the Rogers-Shephard inequalities have been studied deeply and extended to larger classes of measures on R n . For results on the Brunn-Minkowski inequality see [10,11,14,15,18,19,21,22,23,24], and for generalizations of the Rogers-Shephard inequality see [2,3,4,5,13,29].…”

“…Inequality (5) was an important tool in estimating the Banach-Mazur distance between non-symmetric convex bodies and estimating the socalled MM * -estimate for non-symmetric convex bodies (see [6,26] for more details).…”

“…We include their proofs in the Appendix for completeness. For more information on such sets see also [1] For f ∈ LC 0 and 1 ≤ m ≤ n, define the set…”