Learning Convex Sets Under Uniform Distribution

In order to learn a convex set C, an algorithm is given a random sample of points and the information which of the points belong to C. From this sample a set C′ is constructed which is supposed to be a good approximation of C. The algorithm may have a small probability of failing. We measure the quality of the approximation by minimizing the probability that a random test point selected under the same distribution as the sample points is classified correctly. That minimum is taken over a set of distributions associated with C.