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arXiv · 1509.06408

Sections of the regular simplex - Volume formulas and estimates

Hauke Dirksen
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Abstract

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 $n-1$ vertices gives the maximal volume. We show that, for fixed small distances of a hyperplane to the centroid, the hyperplane containing $n-1$ 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.

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Hauke Dirksen. 2016-03-01. Sections of the regular simplex - Volume formulas and estimates. https://doi.org/10.1002/mana201600109

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