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I have a membership oracle to tell me whether a point is inside of some set, S. I would like to find the radius of the largest (origin-centered) hypersphere that is contained in S.

Do you know any good references for this problem? (I'm looking for an algorithm along with a confidence that the hypersphere is fully contained in S).

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Without more knowledge of the structure of $S$, the problem is impossible. For example, if we're inside $\mathbb{R}^n$, and $S = \mathbb{R}^n - B$ for $B$ a countable set of independent Gaussian samples, all oracle queries will say yes but the maximum radius is 0. – Geoffrey Irving Mar 3 '14 at 0:53
Even if S is connected, without further assumptions, the VC dimension of the possible sets is infinite, therefore not learnable from any number of queries. – R B Mar 3 '14 at 6:36

If S is convex, that this is a linear programming problem. It can as such be solved/approximated in polynomial time using the standard sequence of reductions to the ellipsoid algorithm. I do not know any reference that describe it cleanly, but the following book describe it:

Geometric Algorithms and Combinatorial Optimization (Algorithms and Combinatorics) [Paperback] Martin Grötschel, Laszlo Lovasz, Alexander Schrijver

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