Testing Convexity of Discrete Sets in High Dimensions

05/04/2023

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by   Hadley Black, et al.

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We study the problem of testing whether an unknown set S in n dimensions is convex or far from convex, using membership queries. The simplest high-dimensional discrete domain where the problem of testing convexity is non-trivial is the domain {-1,0,1}^n. Our main results are nearly-tight upper and lower bounds of 3^Θ( √(n)) for one-sided error testing of convex sets over this domain with non-adaptive queries. Together with our 3^Ω(n) lower bound on one-sided error testing with samples, this shows that non-adaptive queries are significantly more powerful than samples for this problem.