The maximum number of faces of the minkowski sum of three convex polytopes (original) (raw)

2013, Proceedings of the 29th annual symposium on Symposuim on computational geometry - SoCG '13

We derive tight expressions for the maximum number of k-faces, 0 ≤ k ≤ d − 1, of the Minkowski sum, P 1 + P 2 + P 3 , of three d-dimensional convex polytopes P 1 , P 2 and P 3 , as a function of the number of vertices of the polytopes, for any d ≥ 2. Expressing the Minkowski sum of the three polytopes as a section of their Cayley polytope C, the problem of counting the number of k-faces of P 1 + P 2 + P 3 , reduces to counting the number of (k + 2)-faces of the subset of C comprising of the faces that contain at least one vertex from each P i . 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, where r ≥ d. For d ≥ 4, the maximum values are attained when P 1 , P 2 and P 3 are d-polytopes, whose vertex sets are chosen appropriately from three distinct d-dimensional moment-like curves.

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