Points of continuity of quasiconvex functions on topological vector spaces
type: Item
instance of: scholarly article
zbMATH Open document ID: 1278.26017
author: P. J. Rabier
publication date: 2013-09-30T00:00:00Z
full work available at URL: https://arxiv.org/abs/1206.5775
full work available at URL: jca20040.htm
Mathematics Subject Classification ID: 26B05
Mathematics Subject Classification ID: 52A41
Mathematics Subject Classification ID: 54E52
zbMATH DE Number: 6212965
zbMATH Keywords: quasiconvex function
zbMATH Keywords: topological essential extremum
zbMATH Keywords: point of continuity
zbMATH Keywords: point of discontinuity
zbMATH Keywords: Baire category
zbMATH Keywords: quasicontinuity
MaRDI profile type: Publication
arXiv ID: 1206.5775
arXiv classification: math.OC
arXiv classification: math.GN
title: Points of continuity of quasiconvex functions on topological vector spaces
published in: Journal of Convex Analysis
review text: Let \({(X,\mathcal{T})}\) be a Baire topological vector space and \({f:X\to\overline{\mathbb{R}}}\) be a quasiconvex function, that is \({f(\lambda x+(1-\lambda)y)\leq\max\{f(x),f(y)\}}\) for all \({x,y\in X}\) and every \({\lambda\in[0,1]}.\) In the article, the author investigates continuity properties of such functions. He gives necessary and sufficient conditions for a quasiconvex function \(f\) to be continuous on the residual subset of~\(X,\) the proof of quasicontinuity for any upper semicontinuous quasiconvex function~\(f,\) and other additional statements. The proofs of all facts base on the consideration of an topological essential extrema NEWLINE\[NEWLINE \mathcal{T}\mathrm{ess\,sup}f=\sup\{\alpha\in\mathbb{R}:\{f\mid_U>\alpha\}\;\text{is of second category}\} NEWLINE\]NEWLINE for an open \({U\subset X},\) and rely on some properties of convex sets. Some examples are considered in the article.
reviewed by: Ivan Podvigin
description: scientific article; zbMATH DE number 6212965