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Optimal Volume-Sensitive Bounds for Polytope Approximation

Authors Sunil Arya , David M. Mount

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Author Details

Sunil Arya
  • Department of Computer Science and Engineering, The Hong Kong University of Science and Technology, Hong Kong, China
David M. Mount
  • Department of Computer Science and Institute for Advanced Computer Studies, University of Maryland, College Park, MD, USA

Funding

  • Arya, Sunil: Research supported by the Research Grants Council of Hong Kong, China under project number 16213219 and 16214721.

Acknowledgements

The authors would like to acknowledge the insights and feedback from Rahul Arya and Guilherme da Fonseca.

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Sunil Arya and David M. Mount. Optimal Volume-Sensitive Bounds for Polytope Approximation. In 39th International Symposium on Computational Geometry (SoCG 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 258, pp. 9:1-9:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023) https://doi.org/10.4230/LIPIcs.SoCG.2023.9

Abstract

Approximating convex bodies is a fundamental question in geometry and has a wide variety of applications. Consider a convex body K of diameter Δ in ℝ^d for fixed d. The objective is to minimize the number of vertices (alternatively, the number of facets) of an approximating polytope for a given Hausdorff error ε. It is known from classical results of Dudley (1974) and Bronshteyn and Ivanov (1976) that Θ((Δ/ε)^{(d-1)/2}) vertices (alternatively, facets) are both necessary and sufficient. While this bound is tight in the worst case, that of Euclidean balls, it is far from optimal for skinny convex bodies.
A natural way to characterize a convex object’s skinniness is in terms of its relationship to the Euclidean ball. Given a convex body K, define its volume diameter Δ_d to be the diameter of a Euclidean ball of the same volume as K, and define its surface diameter Δ_{d-1} analogously for surface area. It follows from generalizations of the isoperimetric inequality that Δ ≥ Δ_{d-1} ≥ Δ_d. 
Arya, da Fonseca, and Mount (SoCG 2012) demonstrated that the diameter-based bound could be made surface-area sensitive, improving the above bound to O((Δ_{d-1}/ε)^{(d-1)/2}). In this paper, we strengthen this by proving the existence of an approximation with O((Δ_d/ε)^{(d-1)/2}) facets.
This improvement is a result of the combination of a number of new ideas. As in prior work, we exploit properties of the original body and its polar dual. In order to obtain a volume-sensitive bound, we explore the following more general problem. Given two convex bodies, one nested within the other, find a low-complexity convex polytope that is sandwiched between them. We show that this problem can be reduced to a covering problem involving a natural intermediate body based on the harmonic mean. Our proof relies on a geometric analysis of a relative notion of fatness involving these bodies.

Subject Classification

ACM Subject Classification
  • Theory of computation → Computational geometry
Keywords
  • Approximation algorithms
  • convexity
  • Macbeath regions

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