164 research outputs found

    On f-vectors of Minkowski additions of convex polytopes

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    The objective of this paper is to present two types of results on Minkowski sums of convex polytopes. The first is about a special class of polytopes we call perfectly centered and the combinatorial properties of the Minkowski sum with their own dual. In particular, we have a characterization of face lattice of the sum in terms of the face lattice of a given perfectly centered polytope. Exact face counting formulas are then obtained for perfectly centered simplices and hypercubes. The second type of results concerns tight upper bounds for the f-vectors of Minkowski sums of several polytopes.Comment: 13 pages, submitted to Discrete & Computational Geometr

    f-Vectors of Minkowski Additions of Convex Polytopes

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    The objective of this paper is to present two types of results on Minkowski sums of convex polytopes. The first is about a special class of polytopes we call perfectly centered and the combinatorial properties of the Minkowski sum with their own dual. In particular, we have a characterization of the face lattice of the sum in terms of the face lattice of a given perfectly centered polytope. Exact face counting formulas are then obtained for perfectly centered simplices and hypercubes. The second type of results concerns tight upper bounds for the f-vectors of Minkowski sums of several polytope

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

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    We derive tight expressions for the maximum number of kk-faces, 0kd10\le{}k\le{}d-1, of the Minkowski sum, P1+P2+P3P_1+P_2+P_3, of three dd-dimensional convex polytopes P1P_1, P2P_2 and P3P_3 in Rd\reals^d, as a function of the number of vertices of the polytopes, for any d2d\ge{}2. Expressing the Minkowski sum as a section of the Cayley polytope C\mathcal{C} of its summands, counting the kk-faces of P1+P2+P3P_1+P_2+P_3 reduces to counting the (k+2)(k+2)-faces of C\mathcal{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 rr dd-polytopes in Rd\reals^d, where rdr\ge d. For d4d\ge{}4, the maximum values are attained when P1P_1, P2P_2 and P3P_3 are dd-polytopes, whose vertex sets are chosen appropriately from three distinct dd-dimensional moment-like curves

    Topological obstructions for vertex numbers of Minkowski sums

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    We show that for polytopes P_1, P_2, ..., P_r \subset \R^d, each having n_i \ge d+1 vertices, the Minkowski sum P_1 + P_2 + ... + P_r cannot achieve the maximum of \prod_i n_i vertices if r \ge d. This complements a recent result of Fukuda & Weibel (2006), who show that this is possible for up to d-1 summands. The result is obtained by combining methods from discrete geometry (Gale transforms) and topological combinatorics (van Kampen--type obstructions) as developed in R\"{o}rig, Sanyal, and Ziegler (2007).Comment: 13 pages, 2 figures; Improved exposition and less typos. Construction/example and remarks adde
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