# On Extensions of the Loomis-Whitney Inequality and Ball's Inequality for Concave, Homogeneous Measures

## TL;DR

This paper extends the Loomis-Whitney and Ball inequalities to a broader class of measures that are q-concave and 1/q-homogeneous, broadening their applicability in convex geometry.

## Contribution

It generalizes classical inequalities to q-concave, 1/q-homogeneous measures, providing new bounds for convex bodies under these measures.

## Key findings

- Extended Loomis-Whitney inequality to q-concave measures
- Generalized Ball's inequality for new measure classes
- Provided bounds applicable to a wider range of convex bodies

## Abstract

The Loomis-Whitney inequality states that the volume of a convex body is bounded by the product of volumes of its projections onto orthogonal hyperplanes. We provide an extension of both this fact and a generalization of this fact due to Ball to the context of $q-$concave, $\frac{1}{q}-$homogeneous measures.