On the Width of the Regular n-Simplex

2023/01/06 by Har-Peled, Sariel, Robson, Eliot W.
#Computational Geometry (cs.CG) #FOS: Computer and information sciences

paper · doi:10.48550/arxiv.2301.02616

Abstract

Consider the regular n-simplex Δ_n - it is formed by the convex-hull of n+1 points in Euclidean space, with each pair of points being in distance exactly one from each other. We prove an exact bound on the width of Δ_n which is ≈ √(2/n). Specifically, width(Δ_n) = √((2)/(n + 1)) if n is odd, and width(Δ_n) = √((2(n+1))/(n(n+2))) if n is even. While this bound is well known [GK92, Ale77], we provide a self-contained elementary proof that might (or might not) be of interest.

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