Centroids of sections of convex bodies and Lusternik-Schnirelmann category

Given two symmetric convex bodies $L\subseteq K\subseteq {\mathbb{R}}^n$ with $L$ strictly convex, we prove that there exist at least $n$ hyperplanes $H$ tangent to $L$, such that the center of mass of $H\cap K$ belongs to $\mathrm{\partial }L$. The theorem makes use of Lusternik-Schnirelmann category theory.