On the Width of the Regular n-Simplex

Consider the regular n-simplex Deltan - 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 Deltan which is ≈ √2/n. Specifically, width(Deltan) = √frac2n + 1 if n is odd, and width(Deltan) = √frac2(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.