The Computational Complexity of Convex Bodies, Surveys on Discrete and Computational Geometry
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
Original Citation
Veomett, E. & Barvinok, A. (2008). The computational complexity of convex bodies, surveys on discrete and computational geometry. Contemporary Mathematics, 453, 117-137.
Repository Citation
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
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