On the Width of the Regular $n$-Simplex

Consider the regular $n$-simplex ${\mathrm{\Delta }}_n$ - 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 ${\mathrm{\Delta }}_n$ which is $\approx \sqrt{2\mathrm{/}n}$. Specifically, $\mathrm{w}\mathrm{i}\mathrm{d}\mathrm{t}\mathrm{h}({\mathrm{\Delta }}_n)=\sqrt{\frac{2}{n+1}}$ if $n$ is odd, and $\mathrm{w}\mathrm{i}\mathrm{d}\mathrm{t}\mathrm{h}({\mathrm{\Delta }}_n)=\sqrt{\frac{2(n+1)}{n(n+2)}}$ 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.