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On Testing and Robust Characterizations of Convexity

Authors Eric Blais, Abhinav Bommireddi

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Author Details

Eric Blais
  • University of Waterloo, Canada
Abhinav Bommireddi
  • University of Waterloo, Canada

Acknowledgements

The authors thank Lap Chi Lau for numerous insightful discussions during the course of this research. The authors also thank the anonymous referees for valuable feedback and suggestions.

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Eric Blais and Abhinav Bommireddi. On Testing and Robust Characterizations of Convexity. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 176, pp. 18:1-18:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020) https://doi.org/10.4230/LIPIcs.APPROX/RANDOM.2020.18

Abstract

A body K ⊂ ℝⁿ is convex if and only if the line segment between any two points in K is completely contained within K or, equivalently, if and only if the convex hull of a set of points in K is contained within K. We show that neither of those characterizations of convexity are robust: there are bodies in ℝⁿ that are far from convex - in the sense that the volume of the symmetric difference between the set K and any convex set C is a constant fraction of the volume of K - for which a line segment between two randomly chosen points x,y ∈ K or the convex hull of a random set X of points in K is completely contained within K except with exponentially small probability. These results show that any algorithms for testing convexity based on the natural line segment and convex hull tests have exponential query complexity.

Subject Classification

ACM Subject Classification
  • Theory of computation → Streaming, sublinear and near linear time algorithms
Keywords
  • Convexity
  • Line segment test
  • Convex hull test
  • Intersecting cones

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