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Discrete & Computational Geometry
, Volume 13, Issue 3, pp 541559
First online:
Isoperimetric problems for convex bodies and a localization lemma
 R. KannanAffiliated withDepartment of Computer Science, CarnegieMellon University
 , L. LovászAffiliated withDepartment of Computer Science, Yale University
 , M. SimonovitsAffiliated withMathematical Institute, Hungarian Academy of Sciences
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We study the smallest number ψ(K) such that a given convex bodyK in ℝ^{
n
} can be cut into two partsK
_{1} andK
_{2} by a surface with an (n−1)dimensional measure ψ(K) vol(K
_{1})·vol(K
_{2})/vol(K). LetM
_{1}(K) be the average distance of a point ofK from its center of gravity. We prove for the “isoperimetric coefficient” that, and give other upper and lower bounds. We conjecture that our upper bound is the exact value up to a constant.
$$\psi (K) \geqslant \frac{{\ln 2}}{{M_1 (K)}}$$
Our main tool is a general “Localization Lemma” that reduces integral inequalities over thendimensional 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 wellknown results can be proved using it.
 Title
 Isoperimetric problems for convex bodies and a localization lemma
 Journal

Discrete & Computational Geometry
Volume 13, Issue 34 , pp 541559
 Cover Date
 199506
 DOI
 10.1007/BF02574061
 Print ISSN
 01795376
 Online ISSN
 14320444
 Publisher
 Springer New York
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 Authors

 R. Kannan ^{(1)}
 L. Lovász ^{(2)}
 M. Simonovits ^{(3)}
 Author Affiliations

 1. Department of Computer Science, CarnegieMellon University, 15213, Pittsburgh, PA, USA
 2. Department of Computer Science, Yale University, 06520, New Haven, CT, USA
 3. Mathematical Institute, Hungarian Academy of Sciences, Reáltanoda u. 1315, H1053, Budapest, Hungary