On the theorem of Arrow-Barankin-Blackwell for weakly compact convex sets
type: Item
instance of: scholarly article
title: On the theorem of Arrow-Barankin-Blackwell for weakly compact convex sets
zbMATH Open document ID: 0809.52003
published in: Chinese Annals of Mathematics. Series B
publication date: 1995-03-29T00:00:00Z
review text: Let \(X\) be an ordered real vector space, with positive cone \(C\). Within a compact convex set \(A\), a point which is order-minimal is called efficient, while one minimizing a strictly positive functional is called positive proper efficient. Even in finite-dimensional spaces these are not equivalent; but the Arrow-Barankin-Blackwell theorem states that in \(\mathbb{R}^ n\), with \(C\) the nonnegative orthant, the set of positive proper efficient points of any compact convex body \(A\) is dense in the set of efficient points. This important theorem has previously been generalized to arbitrary real normed spaces, equipped with any ordering for which the positive cone has a base. It has also been generalized (by J. Jahn) to the case in which \(A\) is a weakly compact convex set. In this paper, the author derives a very strong generalization, for weakly compact convex sets in Hausdorff locally convex spaces, in which the positive cone is closed, convex, and has a bounded base.
Mathematics Subject Classification ID: 52A07
zbMATH DE Number: 640437
zbMATH Keywords: Hausdorff locally convex space
zbMATH Keywords: convex cone
zbMATH Keywords: positive proper
zbMATH Keywords: efficient points
zbMATH Keywords: weakly compact convex sets
author: Wantao Fu
reviewed by: Robert Dawson
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 640437