A Proof of the Simplex Mean Width Conjecture

2021/12/06 by Aaron Goldsmith, Goldsmith, Aaron
Computer Science · Mathematics · #52-02 #94-02 #Computational Geometry and Mesh Generation #Digital Image Processing Techniques #FOS: Computer and information sciences #FOS: Mathematics #Information Theory (cs.IT) #Metric Geometry (math.MG) #Point processes and geometric inequalities

paper · pdf · doi:10.48550/arxiv.2112.03393

openalex publication_date 2021/12/06 · openalex created_date 2022/05/05 · openalex updated_date 2026/08/04

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∈\mathbbS^d-1 naturally divide \mathbbS^d-1 into d+1 Voronoi cells and conversely any partition of \mathbbS^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.

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