On the Homothety Conjecture

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}.