Centroids of sections of convex bodies and Lusternik-Schnirelmann category

TL;DR

This paper proves a geometric property involving convex bodies and hyperplanes, utilizing Lusternik-Schnirelmann category theory to establish the existence of certain tangent hyperplanes with specific centroid properties.

Contribution

It introduces a novel application of Lusternik-Schnirelmann category theory to convex geometry, demonstrating the existence of tangent hyperplanes with centroid conditions.

Findings

At least n tangent hyperplanes exist with centroids on the boundary of L.

The result connects convex geometry with topological category theory.

The proof leverages Lusternik-Schnirelmann category to establish geometric existence.

Abstract

Given two symmetric convex bodies $L \subseteq K \subseteq \R^n$ 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 $\partial L$. The theorem makes use of Lusternik-Schnirelmann category theory.