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Diameter, Decomposability, and Minkowski Sums of Polytopes

Part of: Polytopes and polyhedra Graph theory

Published online by Cambridge University Press:  17 December 2018

Antoine Deza  and
Lionel Pournin
Antoine Deza
Affiliation:
Advanced Optimization Laboratory, McMaster University, Hamilton, Ontario, Canada Email: [email protected]
Lionel Pournin
Affiliation:
LIPN, Université Paris 13, Villetaneuse, France Email: [email protected]
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Abstract

We investigate how the Minkowski sum of two polytopes affects their graph and, in particular, their diameter. We show that the diameter of the Minkowski sum is bounded below by the diameter of each summand and above by, roughly, the product between the diameter of one summand and the number of vertices of the other. We also prove that both bounds are sharp. In addition, we obtain a result on polytope decomposability. More precisely, given two polytopes $P$ and $Q$, we show that $P$ can be written as a Minkowski sum with a summand homothetic to $Q$ if and only if $P$ has the same number of vertices as its Minkowski sum with $Q$.

Keywords

Minkowski sumpolytopediameterdecomposability

MSC classification

Primary: 05C12: Distance in graphs
Secondary: 52B11: $n$-dimensional polytopes
Type
Article
Information
Canadian Mathematical Bulletin , Volume 62 , Issue 4 , December 2019 , pp. 741 - 755
Copyright
© Canadian Mathematical Society 2018 

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Footnotes

Antoine Deza is partially supported by the Natural Sciences and Engineering Research Council of Canada Discovery Grant program (RGPIN-2015-06163) and Lionel Pournin by the ANR project SoS (Structures on Surfaces), grant number ANR-17-CE40-0033.

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