Active Learning a Convex Body in Low Dimensions.

Consider a set $P subseteq mathbb{R}^d$ 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 $C$. We show that one can solve this problem in two and three dimensions using $O( mathrm{index}(P) log n)$ queries, where $mathrm{index}(P)$ is the largest subset of points of $P$ in convex position. Furthermore, we show that in two dimensions one can solve this problem using $O( mathrm{price}(P,C) log^2 n )$ oracle queries, where $mathrm{price}(P, C)$ is the minimal number of queries that any algorithm for this specific instance requires.