Centroids of sections of convex bodies and Lusternik-Schnirelmann category

TL;DR

The paper addresses the existence of multiple barycentric-like cuts for symmetric convex bodies by recasting centroids of sections as critical points of a smooth functional on the sphere and applying Lusternik–Schnirelmann category. By defining $V_{K,h}$ and exploiting the strict convexity of $L$, the authors show the functional is even and $C^1$, then deduce at least $n$ pairs of critical points, each corresponding to a hyperplane tangent to $L$ with $c(K∩H) ∈ ∂L$. This yields at least $n$ tangent supporting hyperplanes and, in corollaries, $n$ orthogonal projections for Euclidean balls and related directional centroid-results. The approach highlights a topological variational method that connects LS-category with multiplicity of barycentric-type sections, offering an alternative to Borsuk–Ulam-type arguments in convex geometry.

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.

Figures (1)

• Figure 1: Sets $K_{s,y} \Delta K_{t,x}$ and $K \cap H_{t,x}, K \cap H_{s,y}$.

Theorems & Definitions (8)

• Theorem 4
• Corollary 5
• Corollary 6
• Theorem 7
• Lemma 8
• proof
• Corollary 9
• proof : Proof Theorem \ref{['thm_main']}