A characterization of the ellipsoid through planar grazes

Abstract

In this paper we proved the following: \emph{Let $K, L\subset \mathbb R^3$ be two $O$-symmetric convex bodies with $L\subset \emph{int} K$ strictly convex. Suppose that from every $x$ in $\emph{bd} K$ the graze $\Sigma(L,x)$ is a planar curve and $K$ is almost free with respect to $L$. Then $L$ is an ellipsoid.}