Testing convexity of figures under the uniform distribution

We consider the following basic geometric problem: Given ε ∈ (0, 1/2), a 2-dimensional figure that consists of a black object and a white background is ε-far from convex if it differs in at least an ε fraction of the area from every figure where the black object is convex. How many uniform and independent samples from a figure that is ε-far from convex are needed to detect a violation of convexity with probability at least 2/3? This question arises in the context of designing property testers for convexity. Specifically, a (1-sided error) tester for convexity gets samples from the figure, labeled by their color; it always accepts if the black object is convex; it rejects with probability at least 2/3 if the figure is ε-far from convex. We show that Θ(ε^-4/3) uniform samples are necessary and sufficient for detecting a violation of convexity in an ε-far figure and, equivalently, for testing convexity of figures with 1-sided error. Our testing algorithm runs in time O(ε^-4/3) and thus beats the Ω(ε^-3/2) sample lower bound for learning convex figures under the uniform distribution from [26]. It shows that, with uniform samples, we can check if a set is approximately convex much faster than we can find an approximate representation of a convex set.