A Proof of the Simplex Mean Width Conjecture

TL;DR

This paper proves the longstanding Simplex Mean Width Conjecture, establishing that the regular simplex uniquely maximizes the mean width among all simplices within the unit ball in any dimension.

Contribution

The paper provides a self-contained proof of the SMWC, linking Voronoi partitions of the sphere to the vertices of simplices and demonstrating the optimality of the regular simplex.

Findings

Regular simplex has maximum mean width among all simplices in the unit ball.

Voronoi partitioning of the sphere relates to simplex vertices and their mean width.

Proof applies in arbitrary dimensions, confirming the conjecture universally.

Abstract

The mean width of a convex body is the average distance between parallel supporting hyperplanes when the normal direction is chosen uniformly over the sphere. The Simplex Mean Width Conjecture (SMWC) is a longstanding open problem that says the regular simplex has maximum mean width of all simplices contained in the unit ball and is unique up to isometry. We give a self contained proof of the SMWC in $d$ dimensions. The main idea is that when discussing mean width, $d+1$ vertices $v_i\in\mathbb{S}^{d-1}$ naturally divide $\mathbb{S}^{d-1}$ into $d+1$ Voronoi cells and conversely any partition of $\mathbb{S}^{d-1}$ points to selecting the centroids of regions as vertices. We will show that these two conditions are enough to ensure that a simplex with maximum mean width is a regular simplex.