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.
@InProceedings{blais_et_al:LIPIcs.APPROX/RANDOM.2020.18, author = {Blais, Eric and Bommireddi, Abhinav}, 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 = {Byrka, Jaros{\l}aw and Meka, Raghu}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2020.18}, 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} }
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