Rogers–Shephard and local Loomis–Whitney type inequalities

Abstract

We prove various extensions of the Loomis-Whitney inequality and its dual, where the subspaces on which the projections (or sections) are considered are either spanned by vectors w i of a not necessarily orthonormal basis of R n , or their orthogonal complements. In order to prove such inequalities we estimate the constant in the Brascamp-Lieb inequality in terms of the vectors w i . Restricted and functional versions of the inequality will also be considered.1 k ,

On Rogers–Shephard Type Inequalities for General Measures

“…Although the Rogers-Shephard inequality (1.2) has been recently extended to the functional setting (see e.g. [1,2,12] and the references therein), there seems to be no direct way to derive inequality (1.4) from the above-mentioned functional versions just by considering the function χ K φ, where φ is the density of the given measure, and χ K is the characteristic function of a convex body K (see Remark 2.3). More precisely, in [12, Theorems 4.3 and 4.5], Colesanti extended (1.2) to the more general functional inequality…”

An extension of Berwald's inequality and its relation to Zhang's inequality

“…Our next result is the aforementioned extension of [AAGJV,Lemma 3.3] to p ∈ (−1, ∞). I h is non-increasing, I h (0) = µ(L) = 1 and since h is concave, I h is log-concave (see [AAGJV,Lemma 3.2]).…”

“…is increasing in p ∈ (0, ∞). A famous inequality proved by Berwald [Ber,Satz 7] (see also [AAGJV,Theorem 7.2] for a translation into English) provides a reverse Hölder's inequality for L pnorms (p > 0) of concave functions defined on convex bodies. It states that for any K ∈ K n and any concave function f : K → [0, ∞) , then…”

“…In the context of log-concave functions, the following version of Berwald's inequality (1) on epigraphs of convex functions was proved in [AAGJV,Lemma 3.3]:…”

Rogers-Shephard type inequalities for sections

“…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].…”