(12 pts) Describe efficient separation oracles for each of the following families of convex sets. Here,...

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(12 pts) Describe efficient separation oracles for each of the following families of convex sets. Here, "efficient" means linear time plus O(1) calls to any additional oracles provided to you. (a) The set An B, given separation oracles for A and B. (b) The 1 ball: ||x||1 1. (c) Any convex set A, given a projection oracle for A. Recall that a projection oracle, given a point x, returns == Proj(x) = arg min ||x y||2. (d) The e-neighborhood of a convex set A: given a projection oracle for A. YEA - { | 7y A, xy||z (12 pts) Describe efficient separation oracles for each of the following families of convex sets. Here, "efficient" means linear time plus O(1) calls to any additional oracles provided to you. (a) The set An B, given separation oracles for A and B. (b) The 1 ball: ||x||1 1. (c) Any convex set A, given a projection oracle for A. Recall that a projection oracle, given a point x, returns == Proj(x) = arg min ||x y||2. (d) The e-neighborhood of a convex set A: given a projection oracle for A. YEA - { | 7y A, xy||z