Isoperimetric inequalities for distributions of exponential type (Q1336573)

Let \(E_ 1\) denote the exponential probability distribution with distribution function \(E_ 1 (x) = 1 - \exp (-x)\), \(x \geq 0\), and for any \(n = 1,2,\dots\) consider the product probability measure \(E_ n = E_ 1 \times \dots \times E_ 1\) (\(n\) times). The author is interested in the values of \(E_ n\) on the sets \(A \subset R^ n_ + = [0, \infty)^ n\) having the following property: if \((x_ 1,\dots, x_ n) \in A\), \(y = (y_ 1, \dots, y_ n) \in R^ n_ +\), and \(x_ i \leq y_ i\), \(1 \leq i \leq n\), then \(y \in A\). It is shown that for all \(h \geq 0\) we have \[ E_ n (A + hD_ n) \geq (e^{-h} E^{1/n}_ n (A) + (1 - e^{-h}))^ n,\tag{*} \] where \(D_ n = [0,1]^ n\). An infinite- dimensional version of (*) is also given. Applications of (*) are made to a certain family \(\mathcal F\) of marginal distributions of exponential type and to stochastic processes linearly generated by independent random variables with a common law from \(\mathcal F\).