# $L_p$ functional Busemann-Petty centroid inequality

## TL;DR

This paper establishes a functional version of the $L_p$ Busemann-Petty centroid inequality, extending classical convex geometric inequalities to a broader functional setting with new inequalities for $r$-mixed volumes.

## Contribution

It introduces a functional $r$-mixed volume inequality and derives a new functional form of the $L_p$ Busemann-Petty centroid inequality, expanding the scope of convex geometric analysis.

## Key findings

- Proved inequalities for a class of functional $r$-mixed volumes.
- Established a functional version of the $L_p$ Busemann-Petty centroid inequality.
- Characterized equality cases as centered ellipsoids.

## Abstract

If $K\subset\mathbb{R}^n$ is a convex body and $\Gamma_pK$ is the $p$-centroid body of $K$, the $L_p$ Busemann-Petty centroid inequality states that $\vol(\Gamma_pK) \geq \vol(K)$, with equality if and only if $K$ is an ellipsoid centered at the origin. In this work, we prove inequalities for a type of functional $r$-mixed volume for $1 \leq r < n$, and establish as a consequence, a functional version of the $L_p$ Busemann-Petty centroid inequality. \keywords{Convex body, Moment body, Busemann-Petty centroid} }