# Rogers-Shephard and local Loomis-Whitney type inequalities

## TL;DR

This paper develops functional analogues of classical geometric inequalities like Rogers-Shephard and Loomis-Whitney, providing new inequalities and generalizations that include sharp local reverse versions and intersecting subspace cases.

## Contribution

It introduces novel functional versions of Rogers-Shephard and Loomis-Whitney inequalities, extending their scope and including sharp local reverse inequalities and intersection cases.

## Key findings

- Functional analogues of Rogers-Shephard inequalities derived.
- New sharp local reverse Loomis-Whitney inequalities established.
- Generalizations for intersecting subspaces provided.

## Abstract

We provide functional analogues of the classical geometric inequality of Rogers and Shephard on products of volumes of sections and projections. As a consequence we recover (and obtain some new) functional versions of Rogers-Shephard type inequalities as well as some generalizations of the geometric Rogers-Shephard inequality in the case where the subspaces intersect. These generalizations can be regarded as sharp local reverse Loomis-Whitney inequalities. We also obtain a sharp local Loomis-Whitney inequality.