Rogers-Shephard and local Loomis-Whitney type inequalities

A proof of a Loomis–Whitney type inequality via optimal transport

“…where H r denotes the r-dimensional Hausdorff measure. Clearly, in (1) equality holds when A is a coordinate box. The original proof of (1) by Loomis and Whitney [18] goes back to 1949 and it is based on a discrete approach.…”

On Rogers–Shephard Type Inequalities for General Measures

“…In the particular case of a log-concave integrable function f , this result has been recently obtained in [1,Theorem 1.1].…”

“…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…”

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].…”

“…In view of the Borell-Brascamp Lieb inequality (3), in order to prove inequality (16), it is sufficient only to verify that following inclusion holds:…”