Random weighting, asymptotic counting, and inverse isoperimetry

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.