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.