An integral Jensen inequality for convex multifunctions
type: Item
instance of: scholarly article
title: An integral Jensen inequality for convex multifunctions
zbMATH Open document ID: 0829.26011
DOI: 10.1007/BF03323058
published in: Results in Mathematics
publication date: 1996-01-21T00:00:00Z
review text: 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 \(G\)).
reviewed by: Ioan Raşa
Mathematics Subject Classification ID: 26D15
Mathematics Subject Classification ID: 26B25
Mathematics Subject Classification ID: 26E25
Mathematics Subject Classification ID: 26A51
zbMATH DE Number: 785180
zbMATH Keywords: integral inequality
zbMATH Keywords: convex multifunctions
zbMATH Keywords: Jensen inequality
zbMATH Keywords: set-valued function
zbMATH Keywords: integrable selections
author: Janusz Matkowski
author: Kazimierz Nikodem
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 785180