On the Homothety Conjecture

Abstract

Let $K$ be a convex body in ${\mathbb{R}}^n$ 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 $B_p^n$, $1\le p\le \infty$, 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 Schütt and Werner \cite{SW1994} and Stancu \cite{Stancu2009}.

24 pages, 2 figures

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