A general multidimensional Hermite-Hadamard type inequality
type: Item
instance of: scholarly article
title: A general multidimensional Hermite-Hadamard type inequality
zbMATH Open document ID: 1170.26009
author: Javier Cárcamo
author: Jesús de la Cal
author: L. Escauriaza
published in: Journal of Mathematical Analysis and Applications
publication date: 2009-06-18T00:00:00Z
review text: 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.
reviewed by: Gerald A. Heuer
Mathematics Subject Classification ID: 26D07
zbMATH DE Number: 5566264
zbMATH Keywords: Hermite-Hadamard inequalities
zbMATH Keywords: convex functions
zbMATH Keywords: Brownian motion
zbMATH Keywords: Dirichlet problem
MaRDI profile type: Publication
full work available at URL: j.jmaa.2009.03.044
OpenAlex ID: W2156997635
cites work: Q4321947
cites work: Multidimensional Hermite-Hadamard inequalities and the convex order
cites work: Q4506129
cites work: Q3217388
cites work: Foundations of Modern Probability
cites work: Comparison methods for stochastic models and risks
cites work: The Hermite-Hadamard inequality for convex functions of a vector variable
cites work: Q5559870
DOI: 10.1016/J.JMAA.2009.03.044
description: scientific article; zbMATH DE number 5566264