Multidimensional Hermite–Hadamard inequalities and the convex order

The problem of establishing inequalities of the Hermite–Hadamard type for convex functions on n-dimensional convex bodies translates into the problem of finding appropriate majorants of the involved random vector for the usual convex order. We present two results of partial generality which unify and extend the most part of the multidimensional Hermite–Hadamard inequalities existing in the literature, at the same time that lead to new specific results. The first one fairly applies to the most familiar kinds of polytopes. The second one applies to symmetric random vectors taking values in a closed ball for a given (but arbitrary) norm on Rn. Related questions, such as estimates of approximation and extensions to signed measures, also are briefly discussed.