A characterization of essentially strictly convex functions on reflexive Banach spaces
type: Item
instance of: scholarly article
title: A characterization of essentially strictly convex functions on reflexive Banach spaces
zbMATH Open document ID: 1244.46019
author: M. Volle
author: Jean-Baptiste Hiriart-Urruty
published in: Nonlinear Analysis. Theory, Methods \& Applications. Series A: Theory and Methods
publication date: 2011-12-21T00:00:00Z
review text: Let \(X\) be a reflexive Banach space with dual \(X^*\), let \(J: X\to\mathbb{R}\cup\{\infty\}\) be a properly (not necessarily convex) lower-semicontinuous function with conjugate \(f^*\) and subdifferential mapping \(\partial f\), and let \[ \begin{aligned} MJ(x^*) &:= (\partial f)^{-1}(x^*)\\ &=\text{argmin}\{f(y)-\langle x^*,y\rangle\mid y\in\text{dom\,}f\}\\ &=\{x\in X\mid\langle x^*,y\rangle= f(x)+ f^*(x^*)\}\\ &\subset\partial f^*(x^*).\end{aligned} \] On the basis of the differentiability properties of conjugate functions it is shown that \(MJ\) is single-valued on \(\text{dom}(\partial f^*)\) if and only if \(f\) is essentially strictly convex. The result is specified for functions of the form \(J= f+ h\) where \(f\) is a convex function with \(\text{dom\,}f^*= X^*\) and \(h\) a (not necessarily convex) function with \(\text{dom\,}h^*\neq\emptyset\). If \(f(x)= 1/2\| x\|^2\) then \(MJ= \text{Proc}^f_h\) is the so-called Moreau proximal set-valued mapping associated with \(h\) and \(J^*= (f+ h)^*=\Phi^f_h\) can be viewed as an extension of the Asplund function. Conversely, setting \(h= i_S\) (the indicator function of a set \(S\subset X\)) then an assertion concerning Bregman-Tschebychev sets and functions with respect to the function \(f\) can be derived.
reviewed by: Jörg Thierfelder
Mathematics Subject Classification ID: 46G05
Mathematics Subject Classification ID: 49J52
Mathematics Subject Classification ID: 49J53
Mathematics Subject Classification ID: 26B25
Mathematics Subject Classification ID: 26E15
Mathematics Subject Classification ID: 46N10
zbMATH DE Number: 5991163
zbMATH Keywords: essentially strictly convex function
zbMATH Keywords: essentially smooth function
zbMATH Keywords: subdifferential
zbMATH Keywords: conjugate function
zbMATH Keywords: Asplund function
zbMATH Keywords: proximal set-valued mapping
MaRDI profile type: Publication
full work available at URL: j.na.2011.03.056
OpenAlex ID: W2108190296
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DOI: 10.1016/J.NA.2011.03.056
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description: scientific article; zbMATH DE number 5991163