Active Learning a Convex Body in Low Dimensions

Consider a set P subseteq Red of n points, and a convex body C provided via a separation oracle. The task at hand is to decide for each point of P if it is in C using the fewest number of oracle queries. We show that one can solve this problem in two and three dimensions using O( h(P) log n) queries, where h(P) is the largest subset of points of P in convex position. Furthermore, we show that in two dimensions one can solve this problem using O( v(P,C) log2 n ) oracle queries, where v(P, C) is a lower bound on the minimum number of queries that any algorithm for this specific instance requires.