License: Creative Commons Attribution 3.0 Unported license (CC BY 3.0)
When quoting this document, please refer to the following
DOI: 10.4230/LIPIcs.SoCG.2017.46
URN: urn:nbn:de:0030-drops-71812
URL: http://dagstuhl.sunsite.rwth-aachen.de/volltexte/2017/7181/
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Kerber, Michael ; Tichy, Robert ; Weitzer, Mario

Constrained Triangulations, Volumes of Polytopes, and Unit Equations

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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.

BibTeX - Entry

@InProceedings{kerber_et_al:LIPIcs:2017:7181,
  author =	{Michael Kerber and Robert Tichy and Mario Weitzer},
  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 =	{Boris Aronov and Matthew J. Katz},
  publisher =	{Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{http://drops.dagstuhl.de/opus/volltexte/2017/7181},
  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}
}

Keywords: constrained triangulations, simplotopes, volumes of polytopes, projections of polytopes, unit equations, S-integers
Collection: 33rd International Symposium on Computational Geometry (SoCG 2017)
Issue Date: 2017
Date of publication: 20.06.2017

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