Central slices of the regular simplex (Q1914048)

The following result is proved using probabilistic methods. Let \(S\) be a regular \(n\)-dimensional simplex with edges of length \(\sqrt{2}\). The volume of every \((n -1)\)-dimensional slice of \(S\) passing through the centroid of \(S\) is at most \((\sqrt{n+1})/(\Gamma(n)\sqrt {2})\). This value is attained if and only if the slice contains \(n - 1\) vertices of \(S\).