The Computational Complexity of Convex Bodies, Surveys on Discrete and Computational Geometry
SMC Affiliated Work
School of Science
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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 ∈ 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.
Veomett, E. & Barvinok, A. (2008). The computational complexity of convex bodies, surveys on discrete and computational geometry. Contemporary Mathematics, 453, 117-137.
Veomett, Ellen and Barvinok, Alexander. The Computational Complexity of Convex Bodies, Surveys on Discrete and Computational Geometry (2008). Contemporary Mathematics. 453, 117-137. [article]. https://digitalcommons.stmarys-ca.edu/school-science-faculty-works/266