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DOI: 10.4230/LIPIcs.SoCG.2017.46

Constrained Triangulations, Volumes of Polytopes, and Unit Equations

Authors: Michael Kerber, Robert Tichy, and Mario Weitzer

Published in: LIPIcs, Volume 77, 33rd International Symposium on Computational Geometry (SoCG 2017)

Abstract

Given a polytope P in R^d and a subset U of its vertices, is there a triangulation of 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 P. Our proof relates triangulations of 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 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, and Mario Weitzer. Constrained Triangulations, Volumes of Polytopes, and Unit Equations. In 33rd International Symposium on Computational Geometry (SoCG 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 77, pp. 46:1-46:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)

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@InProceedings{kerber_et_al:LIPIcs.SoCG.2017.46,
  author =	{Kerber, Michael and Tichy, Robert and Weitzer, Mario},
  title =	{{Constrained Triangulations, Volumes of Polytopes, and Unit Equations}},
  booktitle =	{33rd International Symposium on Computational Geometry (SoCG 2017)},
  pages =	{46:1--46:15},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-038-5},
  ISSN =	{1868-8969},
  year =	{2017},
  volume =	{77},
  editor =	{Aronov, Boris and Katz, Matthew J.},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2017.46},
  URN =		{urn:nbn:de:0030-drops-71812},
  doi =		{10.4230/LIPIcs.SoCG.2017.46},
  annote =	{Keywords: constrained triangulations, simplotopes, volumes of polytopes, projections of polytopes, unit equations, S-integers}
}

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Constrained Triangulations, Volumes of Polytopes, and Unit Equations

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