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arXiv · 2606.07163

Restricted Hyperplane Sections of the Cross-Polytope and the Simplex

Silouanos Brazitikos·Christos Pandis
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Abstract

We give a new proof of Webb's theorem on maximal central hyperplane sections of the regular \(n\)-simplex \(\Delta_n\), viewed in its standard embedding in \(\mathbb R^{n+1}\). A similar method also yields sharp maximal estimates for non-central sections of \(\Delta^n\) whose distance \(d\) from the barycenter is small, namely $d< \sqrt{\frac{1}{(n+1)(2n+1)}}.$ Moreover, we obtain sharp volume estimates for central hyperplane sections of the cross-polytope \(B_1^n\) that pass through the barycenter of a facet.

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Silouanos Brazitikos, Christos Pandis. 2026-06-05. Restricted Hyperplane Sections of the Cross-Polytope and the Simplex. https://arxiv.org/abs/2606.07163

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