Testing convexity of figures under the uniform distribution

We consider the following basic geometric problem: Given 𝜖 ∈ (0, 1/2), a 2-dimensional black-and-white figure is ∊- far from convex if it differs in at least an ∊ fraction of the area from every figure where the black object is convex. How many uniform and independent samples from a figure that is ∊- far from convex are needed to detect a violation of convexity with constant probability? This question arises in the context of designing property testers for convexity. We show that Θ(𝜖 ^-4/3 ) uniform samples (and the same running time) are necessary and sufficient for detecting a violation of convexity in an ∊-far figure and, equivalently, for testing convexity of figures with 1-sided error. Our algorithm beats the Ω(𝜖 ^−3∕2 ) lower bound by Schmeltz [32] on the number of samples required for learning convex figures under the uniform distribution. It demonstrates 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.