Affine images of compact convex sets and maximal measures
type: Item
instance of: scholarly article
title: Affine images of compact convex sets and maximal measures
zbMATH Open document ID: 1201.46005
author: Miroslav Kačena
author: J. Spurný
published in: Bulletin des Sciences Mathématiques
publication date: 2009-08-20T00:00:00Z
review text: Let \(X\) and \(Y\) be compact convex subsets of real locally convex Hausdorff topological spaces. Then \(\text{ext\,}X\) denotes the set of extreme points of \(X\). Let \(\varphi: X\to Y\) be a continuous affine mapping, let \({\mathcal M}^1(X)\) denote the set of probability Radon measures on \(X\), and let \(\varphi:{\mathcal M}^1(X)\to{\mathcal M}^1(Y)\) be the mapping induced by \(\varphi\). Then \({\mathcal M}^1_{\max}\) denotes the subset of those measures in \({\mathcal M}^1(X)\) which are maximal or boundary measures. The paper considers the following two questions: {\parindent5mm \begin{itemize}\item[1.] Does the condition \(\varphi(\text{ext\,}X)\subseteq\text{ext\,} Y\) imply that \(\varphi_\sharp({\mathcal M}^1_{\max}(X))\subseteq{\mathcal M}^1_{\max}(Y)\)? \item[2.] Do the conditions \(\varphi(\text{ext\,} X)\subseteq\text{ext\,} Y\), and \(\varphi\) is injective on \(\text{ext\,}X\), imply that \(\varphi_\sharp\) is injective on \({\mathcal M}^1_{\max}(X)\)? \end{itemize}} \textit{S.\,Teleman} [C.\ R.\ Acad.\ Sci., Paris, Sér.\ I 318, No.\,6, 525--528 (1994; Zbl 0830.46006)] stated as a theorem, without proof, that the answer to the first question is `yes', but the present paper gives an example, based on the space \([0,1]\times\{-1,0,1\}\) with a `porcupine' topology of \textit{E.\,Bishop} and \textit{K.\,de Leeuw} [Ann.\ Inst.\ Fourier 9, 305--331 (1959; Zbl 0096.08103)] for which the answer is `no'. Nonetheless, it is proved that the answer is `yes' if \(\text{ext\,}Y\) is a Lindelöf space, and a new proof is given that it is `yes' if \(Y\) is a simplex. It is also proved that the answer to Question 2 is `yes' if \(Y\) is a simplex, but an example is given, again using a porcupine space, for which the amswer is `no'.
reviewed by: Aldric L. Brown
Mathematics Subject Classification ID: 46A55
Mathematics Subject Classification ID: 52A07
zbMATH DE Number: 5597498
zbMATH Keywords: compact convex set
zbMATH Keywords: affine mapping
zbMATH Keywords: maximal measure
zbMATH Keywords: simplex
zbMATH Keywords: face
zbMATH Keywords: porcupine topology
MaRDI profile type: Publication
full work available at URL: j.bulsci.2008.07.002
OpenAlex ID: W1992738035
cites work: Q5611281
cites work: Q3900495
cites work: MAXIMAL MEASURES ON TENSOR PRODUCTS OF COMPACT CONVEX SETS
cites work: Some properties of maximal measures on compact convex sets
cites work: The representations of linear functionals by measures on sets of extreme points
cites work: Q5571891
cites work: Tensor Products, Infinite Products, and Projective Limits of Choquet Simplexes.
cites work: A Weierstrass-Stone theorem for Choquet simplexes
cites work: Q3994935
cites work: Infinite dimensional convexity.
cites work: Q3598957
cites work: Homomorphisms and inverse limits of Choquet simplexes
cites work: Q4772256
cites work: Q5643062
cites work: Lectures on Choquet's theorem
cites work: Convex compact spaces and their maps
cites work: Baire classes of Banach spaces and strongly affine functions
cites work: On Maximal Measures
cites work: Q3317581
DOI: 10.1016/J.BULSCI.2008.07.002
Recommended article: Separable subspaces of affine function spaces on convex compact sets
Recommended article: On Maximal Measures
Recommended article: Topologies and continuous functions on extreme points and pure states
Recommended article: Convex compact spaces and their maps
Recommended article: Q3596466
Recommended article: On convex cones with Schwarz-Zamfirescu property
Recommended article: Q3708534
Recommended article: Convex Corson compacta and Radon measures
Recommended article: The abstract Dirichlet problem for continuous vector-valued functions
Recommended article: A solution of the abstract Dirichlet problem for Baire-one functions
description: scientific article; zbMATH DE number 5597498