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A sharp isoperimetric bound for convex bodies

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Abstract

We consider the problem of lower bounding the Minkowski content of subsets of a convex body with a log-concave probability measure, conditioned on the set size. A bound is given in terms of diameter and set size, which is sharp for all set sizes, dimensions, and norms. In the case of uniform density a stronger theorem is shown which is also sharp.

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References

  1. S. G. Bobkov,Isoperimetric and analytic inequalities for log-concave probability measures, Annals of Probability27 (1999), 1903–1921.

    Article  MATH  MathSciNet  Google Scholar 

  2. B. Bollobás and I. Leader,Edge-isoperimetric inequalities in the grid, Combinatorica11 (1991), 299–314.

    Article  MATH  MathSciNet  Google Scholar 

  3. C. Borell,The Brunn-Minkowski inequality in Gauss space, Inventiones Mathematicae30 (1975), 207–216.

    Article  MATH  MathSciNet  Google Scholar 

  4. C. Borell,Convex set functions in d-space, Periodica Mathematica Hungarica6 (1975), 111–136.

    Article  MathSciNet  Google Scholar 

  5. M. Dyer and A. Frieze,Computing the volume of convex bodies: A case where randomness probably helps, Symposium on Applied Mathematics44 (1991), 123–169.

    MathSciNet  Google Scholar 

  6. R. Kannan, L. Lovász and R. Montenegro,Blocking conductance and mixing in random walks, Combinatorics, Probability and Computing, to appear.

  7. R. Kannan, L. Lovász and M. Simonovits,Isoperimetric problems for convex bodies and a localization lemma, Discrete and Computational Geometry13 (1995), 541–559.

    Article  MATH  MathSciNet  Google Scholar 

  8. L. Lovász and M. Simonovits,Random walks in a convex body and an improved volume algorithm, Random Structures and Algorithms4 (1993), 359–412.

    Article  MATH  MathSciNet  Google Scholar 

  9. V. N. Sudakov and B. S. Cirel’son,Extremal properties of half-spaces for spherically invariant measures (Russian) Problems in the theory of probability distributions, II, Zapiski Nauchnykh Seminarov Leningrad. Otdelenie. Matematicheskii Institut im V. A. Steklova (LOMI)41 (1974), 14–24.

    MathSciNet  Google Scholar 

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Authors and Affiliations

  1. School of Mathematics, Georgia Institute of Technology, 30332-0160, Atlanta, GA, USA

    Ravi Montenegro

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  1. Ravi Montenegro
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Correspondence to Ravi Montenegro.

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Supported in part by VIGRE grants at Yale University and the Georgia Institute of Technology.

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Montenegro, R. A sharp isoperimetric bound for convex bodies. Isr. J. Math. 153, 267–284 (2006). https://doi.org/10.1007/BF02771786

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  • DOI: https://doi.org/10.1007/BF02771786

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