An integral Jensen inequality for convex multifunctions
Let \(X\) and \(Y\) be real Banach spaces and \(D\subset X\) a convex open set. Denote by \(\text{cl} (Y)\) the family of all nonempty closed subsets of \(Y\). A set-valued function \(F: D\to \text{cl} (Y)\) is said to be convex if \(tF(x)+ (1-t) F(y) \subset F(tx+ (1-t) y)\) for all \(x, y\in D\) and \(t\in (0, 1)\). \(F\) is said to be continuous at \(z\in D\) if for every neighbourhood \(V\) of zero in \(Y\) there exists a neighbourhood \(U\) of zero in \(X\) such that \(F(x)\subset F(z)+V\) and \(F(z) \subset F(x)+ V\) for all \(x\in (z+ U) \cap D\). The authors prove that if \(F\) is convex and continuous on \(D\) then for each normalized measure space \((\Omega, \Sigma, \mu)\) and for all \(\mu\)- integrable functions \(\varphi: \Omega\to D\) such that \(\overline {\text{conv}} \varphi (\Omega) \subset D\) we have \(\int_\Omega (F\circ \varphi) d\mu \subset F (\int_\Omega \varphi d\mu)\). (The integral of a multifunction \(G\) is the set of integrals of all integrable selections of \(G\)).
scientific article; zbMATH DE number 785180