Active Learning a Convex Body in Low Dimensions
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Consider a set P ⊆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( index(P) n) queries, where 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( price(P,C) ^2 n ) oracle queries, where price(P, C) is the minimal number of queries that any algorithm for this specific instance requires.
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