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Abstract

We consider the problem of determining whether a given set S in \({\mathbb R}^{n}\) is approximately convex, i.e., if there is a convex set \(K \in {\mathbb R}^{n}\) such that the volume of their symmetric difference is at most ε vol(S) for some given ε. When the set is presented only by a membership oracle and a random oracle, we show that the problem can be solved with high probability using poly(n)(c/ε) n oracle calls and computation time. We complement this result with an exponential lower bound for the natural algorithm that tests convexity along “random” lines. We conjecture that a simple 2-dimensional version of this algorithm has polynomial complexity.

Keywords

Convex Body Random Point Random Oracle Convex Polytope Adjacent Peak 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer-Verlag Berlin Heidelberg 2004

Authors and Affiliations

  • Luis Rademacher
    • 1
  • Santosh Vempala
    • 1
  1. 1.Mathematics Department and CSAILMIT 

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