Isoperimetric problems for convex bodies and a localization lemma
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Our main tool is a general “Localization Lemma” that reduces integral inequalities over then-dimensional space to integral inequalities in a single variable. This lemma was first proved by two of the authors in an earlier paper, but here we give various extensions and variants that make its application smoother. We illustrate the usefulness of the lemma by showing how a number of well-known results can be proved using it.
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- D. Applegate and R. Kannan (1990): Sampling and integration of near log-concave functions,Proc. 23th ACM Symposium on the Theory of Computing, pp. 156–163.Google Scholar
- J. Bokowski and E. Spencer Jr. (1979): Zerlegung Konvexen Körperdurch minimale Trennflächen,J. Reine Angew. Math. 311–312, 80–100.Google Scholar
- J. Bourgain (1991):On the Distribution of Polynomials on High Dimensional Convex Sets, Lecture Notes in Mathematics, Vol. 1469, Springer-Verlag, Berlin, pp. 127–137.Google Scholar
- M. Dyer and A. Frieze (1992): Computing the volume of convex bodies: a case where randomness provably helps, in:Probabilistic Combinatorics and Its Applications (ed. B. Bollobás), Proceedings of Symposia in Applied Mathematics, Vol. 44, American Mathematical Society, Providence, RI, pp. 123–170.CrossRefGoogle Scholar
- M. Dyer, A. Frieze, and R. Kannan (1989): A random polynomial time algorithm for approximating the volume of convex bodies,Proc. 21st ACM Symposium on Theory of Computing, pp. 375–381.Google Scholar
- M. Gromov and V. D. Milman (1984): Brunn theorem and a concentration of volume of convex bodies, GAFA Seminar Notes, Tel Aviv University.Google Scholar
- F. John (1948): Extermum problems with inequalities as subsidiary conditions, in:Studies and Essays Presented to R. Courant, Interscience, New York, pp. 187–204.Google Scholar
- L. Lovász and M. Simonovits (1990): Mixing rate of Markov chains, an isoperimetric inequality, and computing the volume.Proc. 31st IEEE Symposium on Foundations of Computer Science, pp. 346–355.Google Scholar
- V. D. Milman and A. Pajor (1989): Isotropic position and inertia ellipsoids and zonoids of the unit ball of a normedn-dimensional space, in:Geometric Aspects of Functional Analysis (eds. J. Lindenstrauss and V. D. Milman), Lecture Notes in Mathematics, Vol. 1376, Springer-Verlag, Berlin, pp. 64–104.CrossRefGoogle Scholar
- G. Sonnevend (1989): Applications of analytic centers for the numerical solution of semi-infinite, convex programs arising in control theory, DFG Report No. 170/1989, Institut für angew. Mathematik, University of Würzburg.Google Scholar