On the Homothety Conjecture

TL;DR

This paper proves the homothety conjecture for the class of $l_p^n$ unit balls, showing it holds only for the Euclidean ball, and demonstrates the conjecture's validity for general convex bodies when the parameter is sufficiently small.

Contribution

It establishes the homothety conjecture for $l_p^n$ balls only when $p=2$ and extends the conjecture's validity to general convex bodies for small parameters.

Findings

Homothety conjecture holds for $l_p^n$ balls only when $p=2$.

The conjecture is true for general convex bodies when the parameter is small.

Improves previous results by Sch"utt, Werner, and Stancu.

Abstract

Let $K$ be a convex body in $\bbR^n$ and $\d>0$. The homothety conjecture asks: Does $K_{\d}=c K$ imply that $K$ is an ellipsoid? Here $K_{\d}$ is the (convex) floating body and $c$ is a constant depending on $\d$ only. In this paper we prove that the homothety conjecture holds true in the class of the convex bodies $B^n_p$, $1\leq p\leq \infty$, the unit balls of $l_p^n$; namely, we show that $(B^n_p)_{\d} = c B^n_p$ if and only if $p=2$. We also show that the homothety conjecture is true for a general convex body $K$ if $\d$ is small enough. This improvs earlier results by Sch\"utt and Werner \cite{SW1994} and Stancu \cite{Stancu2009}.