# Dual Loomis-Whitney inequalities via information theory

## TL;DR

This paper introduces new information-theoretic bounds on the volume and surface area of geometric bodies, leveraging entropy and a novel $L_1$-Fisher information to relate slices to geometric measures.

## Contribution

It develops a unified framework connecting entropy, Fisher information, and geometric inequalities, including new bounds for convex and polyconvex sets.

## Key findings

- Derived volume bounds for convex bodies using entropy methods.
- Introduced $L_1$-Fisher information with superadditivity properties.
- Established lower bounds on surface area based on slice information.

## Abstract

We establish lower bounds on the volume and the surface area of a geometric body using the size of its slices along different directions. In the first part of the paper, we derive volume bounds for convex bodies using generalized subadditivity properties of entropy combined with entropy bounds for log-concave random variables. In the second part, we investigate a new notion of Fisher information which we call the $L_1$-Fisher information, and show that certain superadditivity properties of the $L_1$-Fisher information lead to lower bounds for the surface areas of polyconvex sets in terms of its slices.