Sections of the regular simplex – Volume formulas and estimates

We state a general formula to compute the volume of the intersection of the regular n‐simplex with some k‐dimensional subspace. It is known that for central hyperplanes the one through the centroid containing vertices gives the maximal volume. We show that, for fixed small distances of a hyperplane to the centroid, the hyperplane containing vertices is still volume maximizing. The proof also yields a new and short argument for the result on central sections. With the same technique we give a partial result for the minimal central hyperplane section. Finally, we obtain a bound for k‐dimensional sections.