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arXiv · 1609.05017

Constrained Triangulations, Volumes of Polytopes, and Unit Equations

Michael Kerber·Robert Tichy·Mario Weitzer
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Abstract

Given a polytope $\mathcal{P}$ in $\mathbb{R}^d$ and a subset $U$ of its vertices, is there a triangulation of $\mathcal{P}$ using $d$-simplices that all contain $U$? We answer this question by proving an equivalent and easy-to-check combinatorial criterion for the facets of $\mathcal{P}$. Our proof relates triangulations of $\mathcal{P}$ to triangulations of its "shadow", a projection to a lower-dimensional space determined by $U$. In particular, we obtain a formula relating the volume of $\mathcal{P}$ with the volume of its shadow. This leads to an exact formula for the volume of a polytope arising in the theory of unit equations.

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Michael Kerber, Robert Tichy, Mario Weitzer. 2016-09-16. Constrained Triangulations, Volumes of Polytopes, and Unit Equations. https://arxiv.org/abs/1609.05017

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