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

SMC Author

Ellen Veomett

SMC Affiliated Work

1

Status

Faculty

School

School of Science

Department

Math/Computer Science

Document Type

Article

Publication Date

2008

Publication / Conference / Sponsorship

Contemporary Mathematics

Description/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 ∈ 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.

Scholarly

yes

Volume

453

First Page

117

Last Page

137

Disciplines

Mathematics

Comments

pdf for October 2006 version of article:

http://www.math.lsa.umich.edu/~barvinok/body.pdf

doi for journal issue, but not article:

http://dx.doi.org/10.1090/conm/453

Original Citation

Veomett, E. & Barvinok, A. (2008). The computational complexity of convex bodies, surveys on discrete and computational geometry. Contemporary Mathematics, 453, 117-137.

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