Planar radial mean bodies are convex

TL;DR

This paper proves that for convex bodies in the plane, their radial mean bodies are convex for all parameters greater than -1, extending known convexity results to negative parameter values.

Contribution

It establishes the convexity of radial mean bodies for convex planar bodies when the parameter p is between -1 and 0, expanding previous results.

Findings

Radial mean bodies are convex for p ≥ 0.

Convexity also holds for p in (-1,0) in 2D.

Extends convexity results to negative parameters.

Abstract

The radial mean bodies of parameter $p>-1$ of a convex body $K \subseteq \mathbb R^n$ are radial sets introduced in [4] by Gardner and Zhang. They are known to be convex for $p\geq 0$. We prove that if $K \subseteq \mathbb R^2$ is a convex body, then its radial mean body of parameter $p$ is convex for every $p \in (-1,0)$.