A characterization of the ellipsoid through planar grazes

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