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.
Consider a set P \subseteq \Re ^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 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) \log^2 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.
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using the fewest number of oracle queries
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a lower bound on the minimum