# A reverse Rogers-Shephard inequality for log-concave functions

## TL;DR

This paper establishes a reverse Rogers-Shephard inequality for log-concave functions, providing new volume estimates for polars of convex bodies and their differences under specific barycenter conditions.

## Contribution

It introduces a reverse inequality for log-concave functions and improves volume estimates for polars of convex bodies with barycenter conditions.

## Key findings

- Proved a reverse Rogers-Shephard inequality for log-concave functions.
- Derived volume estimates for polars of convex bodies.
- Enhanced methods for specific cases of log-concave functions.

## Abstract

We will prove a reverse Rogers-Shephard inequality for log-concave functions. In some particular cases, the method used for general log-concave functions can be slightly improved, allowing us to prove volume estimates for polars of $\ell_p$-diferences of convex bodies whose polar bodies under some condition on the barycenter of their polar bodies.