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The computational complexity of convex bodies

Alexander Barvinok·Ellen Veomett
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Abstract

We discuss how well a given convex body B in a real d-dimensional vector space V can be approximated by a set X for which the membership question: ``given an x in V, does x belong to X?'' can be answered efficiently (in time polynomial in d). We discuss approximations of a convex body by an ellipsoid, by an algebraic hypersurface, by a projection of a polytope with a controlled number of facets, and by a section of the cone of positive semidefinite quadratic forms. We illustrate some of the results on the Traveling Salesman Polytope, an example of a complicated convex body studied in combinatorial optimization.

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Alexander Barvinok, Ellen Veomett. 2006-10-10. The computational complexity of convex bodies. https://arxiv.org/abs/math/0610325

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