The maximum number of faces of the Minkowski sum of two convex polytopes

Authors:
Menelaos I. Karavelas;Eleni Tzanaki
Affiliations:
University of Crete, Heraklion, Greece, and Foundation for Research and Technology - Hellas, Heraklion, Greece;University of Crete, Heraklion, Greece, and Foundation for Research and Technology - Hellas, Heraklion, Greece
Venue:
Proceedings of the twenty-third annual ACM-SIAM symposium on Discrete Algorithms
Year:
2012

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Abstract

We derive tight bounds for the maximum number of k-faces, 0 ≤ k ≤ d − 1, of the Minkowski sum, P1 ⊕ P2, of two d-dimensional convex polytopes P1 and P2, as a function of the number of vertices of the polytopes. For even dimensions d ≥ 2, the maximum values are attained when P1 and P2 are cyclic d-polytopes with disjoint vertex sets. For odd dimensions d ≥ 3, the maximum values are attained when P1 and P2 are [d/2]-neighborly d-polytopes, whose vertex sets are chosen appropriately from two distinct d-dimensional moment-like curves.