Title: On the homothety conjecture

Authors: Elisabeth M. Werner and Deping Ye

Issue: Volume 60 (2011), Issue 1, 1-20


Let $K$ be a convex body in $\mathbb{R}^n$ and $\delta>0$. The homothety conjecture asks: Does $K_{\delta} = c K$ imply that $K$ is an ellipsoid? Here $K_{\delta}$ is the (convex) floating body and $c$ is a constant depending on $\delta$ 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 $\ell_p^n$; namely, we show that $(B^n_p)_{\delta} = cB^n_p$ if and only if $p = 2$. We also show that the homothety conjecture is true for a general convex body $K$ if $\delta$ is small enough. This improves earlier results by Sch\"utt and Werner [C. Sch\"utt and E. Werner, \textit{Homothetic floating bodies}, Geom. Dedicata, \textbf{49} (1994) 335--348] and Stancu [A. Stancu, \textit{Two volume product inequalities and their applications}, Canadian Math. Bulletin, \textbf{52} (2009) 464--472].