On the Width of the Regular n-Simplex

01/06/2023

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by   Sariel Har-Peled, et al.

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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.