Testing Convexity of Discrete Sets in High Dimensions

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 $3^{\widetilde \Theta( \sqrt n)}$ for one-sided error testing of convex sets over this domain with non-adaptive queries. Together with our $3^{\Omega(n)}$ lower bound on one-sided error testing with samples, this shows that non-adaptive queries are significantly more powerful than samples for this problem.