Karavelas, Menelaos I. ;
Tzanaki, Eleni
A Geometric Approach for the Upper Bound Theorem for Minkowski Sums of Convex Polytopes
Abstract
We derive tight expressions for the maximum number of kfaces, k=0,...,d1, of the Minkowski sum, P_1+...+P_r, of r convex dpolytopes 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 hvector calculus, stellar subdivisions and shellings, and generalizes the methodology used in [10] and [9] for proving upper bounds on the fvector 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 kfaces of P_1+...+P_r reduces to bounding the subset of the (k+r1)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 = {8195},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {9783939897835},
ISSN = {18688969},
year = {2015},
volume = {34},
editor = {Lars Arge and J{\'a}nos Pach},
publisher = {Schloss DagstuhlLeibnizZentrum fuer Informatik},
address = {Dagstuhl, Germany},
URL = {http://drops.dagstuhl.de/opus/volltexte/2015/5142},
URN = {urn:nbn:de:0030drops51428},
doi = {10.4230/LIPIcs.SOCG.2015.81},
annote = {Keywords: Convex polytopes, Minkowski sum, upper bound}
}
2015
Keywords: 

Convex polytopes, Minkowski sum, upper bound 
Seminar: 

31st International Symposium on Computational Geometry (SoCG 2015)

Issue date: 

2015 
Date of publication: 

2015 