License: Creative Commons Attribution 3.0 Unported license (CC BY 3.0)
When quoting this document, please refer to the following
DOI: 10.4230/LIPIcs.APPROX/RANDOM.2020.18
URN: urn:nbn:de:0030-drops-126214
URL: https://drops.dagstuhl.de/opus/volltexte/2020/12621/
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Blais, Eric ; Bommireddi, Abhinav

On Testing and Robust Characterizations of Convexity

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LIPIcs-APPROX18.pdf (0.5 MB)

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.

BibTeX - Entry

@InProceedings{blais_et_al:LIPIcs:2020:12621,
  author =	{Eric Blais and Abhinav Bommireddi},
  title =	{{On Testing and Robust Characterizations of Convexity}},
  booktitle =	{Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2020)},
  pages =	{18:1--18:15},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-164-1},
  ISSN =	{1868-8969},
  year =	{2020},
  volume =	{176},
  editor =	{Jaros{\l}aw Byrka and Raghu Meka},
  publisher =	{Schloss Dagstuhl--Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/opus/volltexte/2020/12621},
  URN =		{urn:nbn:de:0030-drops-126214},
  doi =		{10.4230/LIPIcs.APPROX/RANDOM.2020.18},
  annote =	{Keywords: Convexity, Line segment test, Convex hull test, Intersecting cones}
}

Keywords: Convexity, Line segment test, Convex hull test, Intersecting cones
Collection: Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2020)
Issue Date: 2020
Date of publication: 11.08.2020

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