Necessary and sufficient conditions for the validity of Jensen's inequality
Résumé
We consider the d-dimensional Jensen inequality, as it was established by McShane in 1937r. Here T is a functional, φ is a convex function defined on a closed convex set K ⊂ ℝd, and f1,. . ., fd are from some linear space of functions. Our aim is to find necessary and sufficient conditions for the validity of (*). In particular, we show that if we exclude three types of convex sets K, then Jensen's inequality holds for a sublinear functional Tif and only ifT is linear, positive, and satisfies T[1] = 1. Furthermore, for each of the excluded types of convex sets, we present nonlinear, sublinear functionals T for which Jensen's inequality holds. Thus the conditions on K are optimal. Our contributions generalize or complete several known results. © 2013 Springer Basel.