Testing convexity of figures under the uniform distribution

We consider the following basic geometric problem: Given epsilon is an element of(0,1/2), a 2-dimensional black-and-white figure is SMALL ELEMENT OF- far from convex if it differs in at least an SMALL ELEMENT OF fraction of the area from every figure where the black object is convex. How many uniform and independent samples from a figure that is SMALL ELEMENT OF- 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 Theta(epsilon-4/3) uniform samples (and the same running time) are necessary and sufficient for detecting a violation of convexity in an SMALL ELEMENT OF-far figure and, equivalently, for testing convexity of figures with 1-sided error. Our algorithm beats the omega(epsilon-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.