  For a family X of k-subsets of the set 1,...,n, let |X| be the cardinality of
X and let Gamma(X,mu) 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 mu on R. We consider the inverse isoperimetric problem
of finding mu for which Gamma(X,mu) gives the best estimate of ln|X|. We prove
that the optimal choice of mu is the logistic distribution, in which case
Gamma(X,mu) provides an asymptotically tight estimate of ln|X| as k^{-1}ln|X|
grows. Since in many important cases Gamma(X,mu) can be easily computed, we
obtain computationally efficient approximation algorithms for a variety of
counting problems. Given mu, we describe families X of a given cardinality with
the minimum value of Gamma(X,mu), thus extending and sharpening various
isoperimetric inequalities in the Boolean cube.
