scholarly journals On Mosco convergence of convex sets

1988 ◽  
Vol 38 (2) ◽  
pp. 239-253 ◽  
Author(s):  
Gerald Beer
Keyword(s):  
Convex Sets ◽  
Hausdorff Metric ◽  
Mosco Convergence ◽  
Natural Topology ◽  
Weakly Compact ◽  
Reflexive Space ◽  
Convergence Of Sequences ◽  
Metric Topology ◽  
Hausdorff Metric Topology

We present a natural topology compatible with the Mosco convergence of sequences of closed convex sets in a reflexive space, and characterise the topology in terms of the continuity of the distance between convex sets and fixed weakly compact ones. When the space is separable, the topology is Polish. As an application, we show that in this context, most closed convex sets are almost Chebyshev, a result that fails for the stronger Hausdorff metric topology.

scholarly journals Paths in hyperspaces

2003 ◽  
Vol 4 (2) ◽  
pp. 377 ◽  
Author(s):  
Camillo Constantini ◽  
Wieslaw Kubís
Keyword(s):  
Metric Space ◽  
Metric Spaces ◽  
Hausdorff Metric ◽  
Absolute Retract ◽  
Almost Convex ◽  
Convex Metric Spaces ◽  
Metric Topology ◽  
Hausdorff Metric Topology ◽  
Bounded Sets ◽  
Separable Metric Space

<p>We prove that the hyperspace of closed bounded sets with the Hausdor_ topology, over an almost convex metric space, is an absolute retract. Dense subspaces of normed linear spaces are examples of, not necessarily connected, almost convex metric spaces. We give some necessary conditions for the path-wise connectedness of the Hausdorff metric topology on closed bounded sets. Finally, we describe properties of a separable metric space, under which its hyperspace with the Wijsman topology is path-wise connected.</p>

scholarly journals Cofinal completeness of the Hausdorff metric topology

2010 ◽  
Vol 208 (1) ◽  
pp. 75-85 ◽  
Author(s):  
Gerald Beer ◽  
Giuseppe Di Maio
Keyword(s):  
Hausdorff Metric ◽  
Metric Topology ◽  
Hausdorff Metric Topology

scholarly journals The weak drop property on closed convex sets

1994 ◽  
Vol 56 (1) ◽  
pp. 125-130
Author(s):  
Pei-Kee Lin ◽  
Xintai Yu
Keyword(s):  
Convex Sets ◽  
Convex Set ◽  
Nonempty Interior ◽  
Weakly Compact ◽  
Closed Set ◽  
Reflexive Space ◽  
Closed Convex Set

AbstractRecall a closed convex set C is said to have the weak drop property if for every weakly sequentially closed set A disjoint from C there exists x ∈ A such that co({x} ∩ C) ∪ A = {x}. Giles and Kutzarova proved that every bounded closed convex set with the weak drop property is weakly compact. In this article, we show that if C is an unbounded closed convex set of X with the weak drop property, then C has nonempty interior and X is a reflexive space.

Čebyšev Sets in Hyperspaces over Rn

2009 ◽  
Vol 61 (2) ◽  
pp. 299-314 ◽  
Author(s):  
Robert J. MacG. Dawson and ◽  
Maria Moszyńska
Keyword(s):  
Metric Space ◽  
Convex Sets ◽  
Compact Convex ◽  
Hausdorff Metric ◽  
Linear Structure ◽  
Convex Compact ◽  
Nearest Neighbour ◽  
Compact Sets ◽  
Strictly Convex ◽  
Convex Compact Sets

Abstract. A set in a metric space is called a Čebyšev set if it has a unique “nearest neighbour” to each point of the space. In this paper we generalize this notion, defining a set to be Čebyšev relative to another set if every point in the second set has a unique “nearest neighbour” in the first. We are interested in Čebyšev sets in some hyperspaces over Rn, endowed with the Hausdorff metric, mainly the hyperspaces of compact sets, compact convex sets, and strictly convex compact sets. We present some new classes of Čebyšev and relatively Čebyšev sets in various hyperspaces. In particular, we show that certain nested families of sets are Čebyšev. As these families are characterized purely in terms of containment,without reference to the semi-linear structure of the underlyingmetric space, their properties differ markedly from those of known Čebyšev sets.

scholarly journals Properties of the trajectories of set-valued integrals in banach spaces

1990 ◽  
Vol 41 (2) ◽  
pp. 271-281
Author(s):  
Nikolaos S. Papageorgiou
Keyword(s):  
Banach Space ◽  
Banach Spaces ◽  
Separable Banach Space ◽  
Hausdorff Metric ◽  
Density Theorem ◽  
Mosco Convergence ◽  
Convexity Theorem ◽  
Convergence Theorems ◽  
Convergence Of Sets ◽  
Differentiability Properties

Let F: T → 2x \ {} be a closed-valued multifunction into a separable Banach space X. We define the sets and We prove various convergence theorems for those two sets using the Hausdorff metric and the Kuratowski-Mosco convergence of sets. Then we prove a density theorem of CF and a corresponding convexity theorem for F(·). Finally we study the “differentiability” properties of K(·). Our work extends and improves earlier ones by Artstein, Bridgland, Hermes and Papageorgiou.

The Demyanov metric and some other metrics in the family of convex sets

2012 ◽  
Vol 10 (6) ◽  
Author(s):  
Tadeusz Rzeżuchowski
Keyword(s):  
Convex Sets ◽  
Hausdorff Metric ◽  
Facial Structure ◽  
The Family

AbstractWe describe some known metrics in the family of convex sets which are stronger than the Hausdorff metric and propose a new one. These stronger metrics preserve in some sense the facial structure of convex sets under small changes of sets.

scholarly journals Fixed points of isometries on weakly compact convex sets

2003 ◽  
Vol 282 (1) ◽  
pp. 1-7 ◽  
Author(s):  
Teck-Cheong Lim ◽  
Pei-Kee Lin ◽  
C. Petalas ◽  
T. Vidalis
Keyword(s):  
Fixed Points ◽  
Convex Sets ◽  
Compact Convex ◽  
Weakly Compact

scholarly journals On Fixed Point Property under Lipschitz and Uniform Embeddings

2018 ◽  
Vol 2018 ◽  
pp. 1-6 ◽  
Author(s):  
Jichao Zhang ◽  
Lingxin Bao ◽  
Lili Su
Keyword(s):  
Banach Space ◽  
Fixed Point ◽  
Convex Subset ◽  
Convex Sets ◽  
Fixed Point Property ◽  
Point Property ◽  
Lipschitz Mappings ◽  
Affine Mappings ◽  
Reflexive Space ◽  
Gateaux Differentiability

We first present a generalization of ω⁎-Gâteaux differentiability theorems of Lipschitz mappings from open sets to those closed convex sets admitting nonsupport points and then show that every nonempty bounded closed convex subset of a Banach space has the fixed point property for isometries if it Lipschitz embeds into a super reflexive space. With the application of Baudier-Lancien-Schlumprecht’s theorem, we finally show that every nonempty bounded closed convex subset of a Banach space has the fixed point property for continuous affine mappings if it uniformly embeds into the Tsirelson space T⁎.

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