A lemma on convex functionals in finite-dimensional spaces
type: Item
instance of: scholarly article
title: A lemma on convex functionals in finite-dimensional spaces
zbMATH Open document ID: 0641.49008
DOI: 10.1007/BF01194578
author: Jochen W. Schmidt
author: Wolfgang Burmeister
published in: Archiv der Mathematik
publication date: 1988-01-01T00:00:00Z
review text: The authors prove the lemma below, then describe its applications to the subdifferentiability and cone radius of convex operators. (A convex operator is a mapping \(L: D\subset {\mathbb{R}}^ n\to {\mathbb{R}}^ n\) which is convex with respect to the ordering induced by a closed convex subcone of \(R^ n\) with vertex at 0.) Lemma: Let D be a convex subset of \({\mathbb{R}}^ n \)with nonempty interior, and let \(f_ 1,...,f_ s:D\to {\mathbb{R}}\) be convex functionals. Suppose that for some points \(x_ 0\in int D\) and \(x_ 1,...,x_ s\in D\), the map \(x\mapsto \sum^{s}_{i=1}f_ i(x)(x_ i-x_ 0)\) is constant on D. Then one has the inequality \(\sum^{s}_{i=1}f_ i(x_ i)\geq \sum^{s}_{i=1}f_ i(x_ 0)\).
Mathematics Subject Classification ID: 49J52
Mathematics Subject Classification ID: 52A40
Mathematics Subject Classification ID: 90C25
Mathematics Subject Classification ID: 26B05
Mathematics Subject Classification ID: 26B25
zbMATH DE Number: 4044579
zbMATH Keywords: subdifferentiability
zbMATH Keywords: cone radius
zbMATH Keywords: convex operators
reviewed by: P. D. Loewen
MaRDI profile type: Publication
Wikidata QID: Q124886881
cites work: Determination of the cone radius for positive concave operators
cites work: Q4104377
cites work: Q5558820
cites work: Q5603731
cites work: Subdifferentiability of Convex Functions with Values in an Ordered Vector Space.
full work available at URL: https://doi.org/10.1007/bf01194578
OpenAlex ID: W2007626532
description: scientific article; zbMATH DE number 4044579