The Computational Complexity of Convex Bodies, Surveys on Discrete and Computational Geometry

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.