# Random weighting, asymptotic counting, and inverse isoperimetry

- Received:

DOI: 10.1007/s11856-007-0008-8

- Cite this article as:
- Barvinok, A. & Samorodnitsky, A. Isr. J. Math. (2007) 158: 159. doi:10.1007/s11856-007-0008-8

- 12 Citations
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## Abstract

For a family *X* of *k*-subsets of the set {1, …, *n*}, let |*X*| be the cardinality of *X* and let Γ(*X, μ*) be the expected maximum weight of a subset from *X* when the weights of 1, …, *n* are chosen independently at random from a symmetric probability distribution *μ* on ℝ. We consider the inverse isoperimetric problem of finding *μ* for which Γ(*X, μ*) gives the best estimate of ln |*X*|. We prove that the optimal choice of *μ* is the logistic distribution, in which case Γ(*X, μ*) provides an asymptotically tight estimate of ln |*X*| as *k*^{−1} ln |*X*} grows. Since in many important cases Γ(*X, μ*) can be easily computed, we obtain computationally efficient approximation algorithms for a variety of counting problems. Given *μ*, we describe families *X* of a given cardinality with the minimum value of Γ(*X, μ*), thus extending and sharpening various isoperimetric inequalities in the Boolean cube.