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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.2015.81
URN: urn:nbn:de:0030-drops-51428
URL: http://dagstuhl.sunsite.rwth-aachen.de/volltexte/2015/5142/
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Karavelas, Menelaos I. ; Tzanaki, Eleni

A Geometric Approach for the Upper Bound Theorem for Minkowski Sums of Convex Polytopes

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Abstract

We derive tight expressions for the maximum number of k-faces, k=0,...,d-1, of the Minkowski sum, P_1+...+P_r, of r convex d-polytopes P_1,...,P_r in R^d, where d >= 2 and r < d, as a (recursively defined) function on the number of vertices of the polytopes. Our results coincide with those recently proved by Adiprasito and Sanyal [1]. In contrast to Adiprasito and Sanyal's approach, which uses tools from Combinatorial Commutative Algebra, our approach is purely geometric and uses basic notions such as f- and h-vector calculus, stellar subdivisions and shellings, and generalizes the methodology used in [10] and [9] for proving upper bounds on the f-vector of the Minkowski sum of two and three convex polytopes, respectively. The key idea behind our approach is to express the Minkowski sum P_1+...+P_r as a section of the Cayley polytope C of the summands; bounding the k-faces of P_1+...+P_r reduces to bounding the subset of the (k+r-1)-faces of C that contain vertices from each of the r polytopes. We end our paper with a sketch of an explicit construction that establishes the tightness of the upper bounds.

BibTeX - Entry

@InProceedings{karavelas_et_al:LIPIcs:2015:5142,
  author =	{Menelaos I. Karavelas and Eleni Tzanaki},
  title =	{{A Geometric Approach for the Upper Bound Theorem for Minkowski Sums of Convex Polytopes}},
  booktitle =	{31st International Symposium on Computational Geometry (SoCG 2015)},
  pages =	{81--95},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-939897-83-5},
  ISSN =	{1868-8969},
  year =	{2015},
  volume =	{34},
  editor =	{Lars Arge and J{\'a}nos Pach},
  publisher =	{Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{http://drops.dagstuhl.de/opus/volltexte/2015/5142},
  URN =		{urn:nbn:de:0030-drops-51428},
  doi =		{10.4230/LIPIcs.SOCG.2015.81},
  annote =	{Keywords: Convex polytopes, Minkowski sum, upper bound}
}

Keywords: Convex polytopes, Minkowski sum, upper bound
Collection: 31st International Symposium on Computational Geometry (SoCG 2015)
Issue Date: 2015
Date of publication: 12.06.2015

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