The maximum number of faces of the minkowski sum of three convex polytopes

  • Authors:

  • Menelaos I. Karavelas;Christos Konaxis;Eleni Tzanaki

  • Affiliations:

  • University of Crete & Foundation for Research and Technology - Helas, Heraklion, Greece;University of Crete, Heraklion, Greece;University of Crete & Foundation for Research and Technology - Helas, Heraklion, Greece

  • Venue:
  • Proceedings of the twenty-ninth annual symposium on Computational geometry
  • Year:
  • 2013

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Abstract

We derive tight expressions for the maximum number of k-faces, 0≤k≤d-1, of the Minkowski sum, P1+P2+P3, of three d-dimensional convex polytopes P1, P2 and P3 in Rd, as a function of the number of vertices of the polytopes, for any d≥2. Expressing the Minkowski sum as a section of the Cayley polytope C of its summands, counting the k-faces of P1+P2+P3 reduces to counting the (k+2)-faces of C which meet the vertex sets of the three polytopes. In two dimensions our expressions reduce to known results, while in three dimensions, the tightness of our bounds follows by exploiting known tight bounds for the number of faces of r d-polytopes in Rd, where r≥d. For d≥4, the maximum values are attained when P1, P2 and P3 are d-polytopes, whose vertex sets are chosen appropriately from three distinct d-dimensional moment-like curves.