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Primitive Zonotopes

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Abstract

We introduce and study a family of polytopes which can be seen as a generalization of the permutahedron of type \(B_d\). We highlight connections with the largest possible diameter of the convex hull of a set of points in dimension d whose coordinates are integers between 0 and k, and with the computational complexity of multicriteria matroid optimization.

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Acknowledgements

The authors thank the anonymous referees, Johanne Cohen, Nathann Cohen, Komei Fukuda, and Aladin Virmaux for valuable comments and for informing us of reference [25], Emo Welzl and Günter Ziegler for helping us access Thorsten Thiele’s Diplomarbeit, Dmitrii Pasechnik for pointing out reference [23] and the concept of Minkowski length, and Vincent Pilaud for pointing out graphical zonotopes and that \(Z_1(d,2)\) is the permutahedron of type \(B_d\). This work was partially supported by the Natural Sciences and Engineering Research Council of Canada Discovery Grant program (RGPIN-2015-06163), by the Digiteo Chair C&O program, and by the Dresner Chair at the Technion.

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Authors and Affiliations

  1. Advanced Optimization Laboratory, Faculty of Engineering, McMaster University, Hamilton, ON, L8S 4K1, Canada

    Antoine Deza

  2. Laboratoire de Recherche en Informatique, Université de Paris Sud, 91405, Orsay, France

    George Manoussakis

  3. Operations Research, The William Davidson Faculty of IE & M, Technion Israel Institute of Technology, 3200003, Haifa, Israel

    Shmuel Onn

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  1. Antoine Deza
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  2. George Manoussakis
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  3. Shmuel Onn
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Correspondence to Antoine Deza.

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Editor in Charge: Günter M. Ziegler

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Deza, A., Manoussakis, G. & Onn, S. Primitive Zonotopes. Discrete Comput Geom 60, 27–39 (2018). https://doi.org/10.1007/s00454-017-9873-z

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