Rogers-Shephard type inequalities for sections

Abstract

In this paper we address the following question: given a measure µ on R n , does there exists a constant C > 0 such that, for any m-dimensional subspace H ⊂ R n and any convex body K ⊂ R n , the following sectional Rogers-Shephard type inequality holds:We show that this inequality is affirmative in the class of measures with radially decreasing densities with the constant C(n, m) = n+m m . We also prove marginal inequalities of the Rogers-Shephard type for 1 s -concave, 0 ≤ s < ∞, and logarithmically concave functions.

On Rogers–Shephard-type inequalities for the lattice point enumerator

“…The Rogers-Shephard inequality was recently extended to the functional setting [4,6,9,14], generalized to different types of measures [8,31], as well as studied in the L p setting [3,13]. Moreover, it was recently extended to other geometric functionals [7], and a reverse form of Rogers-Shephard inequality in the setting of log-concave functions was given in [3].…”

General Measure Extensions of Projection Bodies

“…This result is part of a class of reverse Rogers-Shepard type inequalities, examined further in [1,2,5,45].…”

“…This seems restrictive, but this requirement is necessary to obtain a generalization of Zhang's inequality that reduces to (1) in the case of the Lebesgue measure. We start with a crucial lemma inspired by [45]. Lemma 3.2.…”

General measure extensions of projection bodies

“…The following Rogers-Shephard inequality for radially decreasing measures was proved in [5]: This result is part of a collection of Rogers-Shephard and reverse Rogers-Shephard type inequalities, examined further in [1,2,4,17,55]. In Section 3, we discuss analogues of Zhang's inequality for measures with positive density.…”