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Convex cones, integral zonotopes, limit shape

Barany, I; Bureaux, J; Lund, B; (2018) Convex cones, integral zonotopes, limit shape. Advances in Mathematics , 331 pp. 143-169. 10.1016/j.aim.2018.03.031. Green open access

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Abstract

Given a convex cone C in R d , an integral zonotope T is the sum of segments [0, vi] (i = 1, . . . , m) where each vi ∈ C is a vector with integer coordinates. The endpoint of T is k = Pm 1 vi. Let T (C, k) be the family of all integral zonotopes in C whose endpoint is k ∈ C. We prove that, for large k, the zonotopes in T (C, k) have a limit shape, meaning that, after suitable scaling, the overwhelming majority of the zonotopes in T (C, k) are very close to a fixed convex set. We also establish several combinatorial properties of a typical zonotope in T (C, k).

Type: Article
Title: Convex cones, integral zonotopes, limit shape
Open access status: An open access version is available from UCL Discovery
DOI: 10.1016/j.aim.2018.03.031
Publisher version: http://doi.org/10.1016/j.aim.2018.03.031
Language: English
Additional information: This version is the author accepted manuscript. For information on re-use, please refer to the publisher’s terms and conditions.
Keywords: Convex cones, Integral zonotopes, Integer partitions, Limit shape
UCL classification: UCL
UCL > Provost and Vice Provost Offices
UCL > Provost and Vice Provost Offices > UCL BEAMS
UCL > Provost and Vice Provost Offices > UCL BEAMS > Faculty of Maths and Physical Sciences
UCL > Provost and Vice Provost Offices > UCL BEAMS > Faculty of Maths and Physical Sciences > Dept of Mathematics
URI: https://discovery.ucl.ac.uk/id/eprint/1522761
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