# A Convex Body Associated to the Busemann Random Simplex Inequality and the Petty conjecture

## TL;DR

This paper introduces a new convex body associated with the Busemann Random Simplex Inequality, providing a unified approach to related inequalities and offering insights into the Petty conjecture through dual and functional perspectives.

## Contribution

It defines the convex body $N_p L$ and proves an isoperimetric inequality linked to the Busemann Random Simplex Inequality, also establishing dual and functional versions related to Petty's conjecture.

## Key findings

- Proved an isoperimetric inequality for $(N_p L)^ imes$ equivalent to the Busemann Random Simplex Inequality.
- Provided a simple proof of a functional version of the inequality.
- Established the Petty conjecture's equivalence to an $L_1$-Sharp Affine Sobolev-type inequality.

## Abstract

Given $L$ a convex body, the $L_p$-Busemann Random Simplex Inequality is closely related to the centroid body $\Gamma_p L$ for $p=1$ and $2$, and only in these cases it can be proved using the $L_p$-Busemann-Petty centroid inequality. We define a convex body $N_p L$ and prove an isoperimetric inequality for $(N_p L)^\circ$ that is equivalent to the $L_p$-Busemann Random Simplex Inequality. As applications, we give a simple proof of a general functional version of the Busemann Random Simplex Inequality and study a dual theory related to Petty's conjectured inequality. More precisely, we prove dual versions of the $L_p$-Busemann Random Simplex Inequality for sets and functions by means of the $p$-affine surface area measure, and we prove that the Petty conjecture is equivalent to an $L_1$-Sharp Affine Sobolev-type inequality that is stronger than (and directly implies) the Sobolev-Zhang inequality.