Lower Bounds for Convexity Testing

We consider the problem of testing whether an unknown and arbitrary set $S \subseteq \mathbb{R}^n$ (given as a black-box membership oracle) is convex, versus $\varepsilon$-far from every convex set, under the standard Gaussian distribution. The current state-of-the-art testing algorithms for this problem make $2^{\tilde{O}(\sqrt{n})\cdot \mathrm{poly}(1/\varepsilon)}$ non-adaptive queries, both for the standard testing problem and for tolerant testing. We give the first lower bounds for convexity testing in the black-box query model: - We show that any one-sided tester (which may be adaptive) must use at least $n^{Ω(1)}$ queries in order to test to some constant accuracy $\varepsilon>0$. - We show that any non-adaptive tolerant tester (which may make two-sided errors) must use at least $2^{Ω(n^{1/4})}$ queries to distinguish sets that are $\varepsilon_1$-close to convex versus $\varepsilon_2$-far from convex, for some absolute constants $0<\varepsilon_1<\varepsilon_2$. Finally, we also show that for any constant $c>0$, any non-adaptive tester (which may make two-sided errors) must use at least $n^{1/4 - c}$ queries in order to test to some constant accuracy $\varepsilon>0$.