A Proof of the Simplex Mean Width Conjecture

12/06/2021

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by   Aaron Goldsmith, et al.

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