A general multidimensional Hermite-Hadamard type inequality
The classical Hermite-Hadamard inequality states that for a real convex function \(f\) on an interval \([a,b]\), \[ f\biggl({a+b\over2}\biggr)\leq{1\over b-a}\int_a^b f(x)dx\leq{f(a)+f(b)\over2}. \] This may be expressed in probabilistic terms in the form \[ f(E\xi)\leq Ef(\xi)\leq Ef(\xi^*), f\in C_{cx},\eqno(1) \] where \(E\) denotes expected value, \(\xi\) and \(\xi^*\) are random variables with uniform distributions on the interval \([a,b]\) and the set \(\{a,b\}\), respectively, and \(C_{cx}\) is the set of all real convex functions on \([a,b]\). In this work the authors study multidimensional analogs where \([a,b]\) is replaced by a nonempty compact convex set \(K\subset R^d (d\geq2)\), \(\xi\) is an arbitrary random vector taking values in \(K\), and \(C_{cx}\) is the set of all real continuous convex functions on \(K\). The first inequality in (1) continues to hold by Jensen's inequality. Strong and weak problems related to the second inequality are described. Using a stochastic approach the authors deal with the weak problem, which is to find an \(H_*\)-majorant of \(\xi\), that is, (the probability distribution of) a random vector \(\xi_*\) taking values in \(K_*\), the boundary of \(K\), such that \(Ef(\xi)\leq Ef(\xi_*)\). A connection with the Dirichlet problem is discussed.
scientific article; zbMATH DE number 5566264