scholarly journals The extreme points of some convex sets in the theory of majorization

1987 ◽  
Vol 90 (2) ◽  
pp. 171-176
Author(s):  
Anthony Horsley ◽  
Andrzej J. Wrobel
Keyword(s):  
Convex Sets ◽  
Extreme Points

Extreme points of convex sets

1973 ◽  
Vol 205 (4) ◽  
pp. 299-302
Author(s):  
Walter Pranger
Keyword(s):  
Convex Sets ◽  
Extreme Points

On the extreme points of the sum of two compact convex sets

1970 ◽  
Vol 188 (2) ◽  
pp. 113-122 ◽  
Author(s):  
T. Husain ◽  
I. Tweddle
Keyword(s):  
Convex Sets ◽  
Compact Convex ◽  
Extreme Points

scholarly journals Linear metric spaces and analytic sets

1994 ◽  
Vol 37 (2) ◽  
pp. 355-358
Author(s):  
Robert Kaufman
Keyword(s):  
Set Theory ◽  
Convex Sets ◽  
Metric Spaces ◽  
Compact Convex ◽  
Extreme Points ◽  
Descriptive Set Theory ◽  
Analytic Sets ◽  
Descriptive Set

A problem in descriptive set theory, in which the objects of interest are compact convex sets in linear metric spaces, primarily those having extreme points.

scholarly journals Approximating Extreme Points of Infinite Dimensional Convex Sets

1998 ◽  
Vol 23 (2) ◽  
pp. 433-442 ◽  
Author(s):  
William P. Cross ◽  
H. Edwin Romeijn ◽  
Robert L. Smith
Keyword(s):  
Convex Sets ◽  
Extreme Points ◽  
Infinite Dimensional

scholarly journals Mazur's intersection property of balls for compact convex sets

1987 ◽  
Vol 35 (2) ◽  
pp. 267-274 ◽  
Author(s):  
J. H. M. Whitfield ◽  
V. Zizler
Keyword(s):  
Banach Space ◽  
Uniform Convergence ◽  
Convex Sets ◽  
Convex Set ◽  
Compact Convex ◽  
Extreme Points ◽  
Intersection Property ◽  
Compact Sets ◽  
Compact Convex Set ◽  
Mazur's Intersection Property

We show that every compact convex set in a Banach space X is an intersection of balls provided the cone generated by the set of all extreme points of the dual unit ball of X* is dense in X* in the topology of uniform convergence on compact sets in X. This allows us to renorm every Banach space with transfinite Schauder basis by a norm which shares the mentioned intersection property.

scholarly journals Strongly exposed points in bases for the positive cone of ordered Banach spaces and characterizations of l1(Г)

1986 ◽  
Vol 29 (2) ◽  
pp. 271-282 ◽  
Author(s):  
Ioannis A. Polyrakis
Keyword(s):  
Banach Space ◽  
Banach Spaces ◽  
Convex Sets ◽  
Extreme Points ◽  
Positive Cone ◽  
Infinite Dimensional ◽  
Strongly Exposed Points ◽  
Nikodym Property ◽  
Choquet Theory ◽  
Ordered Banach Spaces

The study of extreme, strongly exposed points of closed, convex and bounded sets in Banach spaces has been developed especially by the interconnection of the Radon–Nikodým property with the geometry of closed, convex and bounded subsets of Banach spaces [5],[2] . In the theory of ordered Banach spaces as well as in the Choquet theory, [4], we are interested in the study of a special type of convex sets, not necessarily bounded, namely the bases for the positive cone. In [7] the geometry (extreme points, dentability) of closed and convex subsets K of a Banach space X with the Radon-Nikodým property is studied and special emphasis has been given to the case where K is a base for acone P of X. In [6, Theorem 1], it is proved that an infinite-dimensional, separable, locally solid lattice Banach space is order-isomorphic to l1 if, and only if, X has the Krein–Milman property and its positive cone has a bounded base.

The Structure of C*-Convex Sets

1994 ◽  
Vol 46 (5) ◽  
pp. 1007-1026 ◽  
Author(s):  
Phillip B. Morenz
Keyword(s):  
Convex Sets ◽  
Extreme Points ◽  
Structural Elements ◽  
The Right ◽  
The Relationship ◽  
Carathéodory Theorem ◽  
Convex Subsets

AbstractCompact C*-convex subsets of Mn correspond exactly to n-th matrix ranges of operators. The main result of this paper is to discover the “right” analog of linear extreme points, called structural elements, and then to prove a generalised Krein-Milman theorem for C*-convex subsets of Mn. The relationship between structural elements and an earlier attempted generalisation, called C*-extreme points, is examined, solving affirmatively a conjecture of Loebl and Paulsen [8]. An improved bound for a C* -convex version of the Caratheodory theorem for convex sets is also given.

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