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Economical Delone Sets for Approximating Convex Bodies

Authors Ahmed Abdelkader , David M. Mount

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Ahmed Abdelkader
  • Department of Computer Science , University of Maryland, College Park MD, USA
David M. Mount
  • Department of Computer Science and Institute of Advanced Computer Studies , University of Maryland, College Park MD, USA

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Ahmed Abdelkader and David M. Mount. Economical Delone Sets for Approximating Convex Bodies. In 16th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 101, pp. 4:1-4:12, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2018)


Convex bodies are ubiquitous in computational geometry and optimization theory. The high combinatorial complexity of multidimensional convex polytopes has motivated the development of algorithms and data structures for approximate representations. This paper demonstrates an intriguing connection between convex approximation and the classical concept of Delone sets from the theory of metric spaces. It shows that with the help of a classical structure from convexity theory, called a Macbeath region, it is possible to construct an epsilon-approximation of any convex body as the union of O(1/epsilon^{(d-1)/2}) ellipsoids, where the center points of these ellipsoids form a Delone set in the Hilbert metric associated with the convex body. Furthermore, a hierarchy of such approximations yields a data structure that answers epsilon-approximate polytope membership queries in O(log (1/epsilon)) time. This matches the best asymptotic results for this problem, by a data structure that both is simpler and arguably more elegant.

Subject Classification

ACM Subject Classification
  • Theory of computation → Computational geometry
  • Approximate polytope membership
  • Macbeath regions
  • Delone sets
  • Hilbert geometry


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