Extreme Points of Convex Sets in Infinite Dimensional Spaces

1987 ◽  
Vol 94 (5) ◽  
pp. 409 ◽  
Author(s):  
Nina M. Roy
Keyword(s):  
Convex Sets ◽  
Extreme Points ◽  
Infinite Dimensional

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 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.

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 Valuations on convex functions and convex sets and Monge–Ampère operators

2019 ◽  
Vol 19 (3) ◽  
pp. 313-322 ◽  
Author(s):  
Semyon Alesker
Keyword(s):  
Convex Functions ◽  
Convex Sets ◽  
Convex Bodies ◽  
Linear Functionals ◽  
Translation Invariant ◽  
Infinite Dimensional ◽  
Dense Image ◽  
Monge Ampere ◽  
Explicit Relation ◽  
Class Of Functions

Abstract The notion of a valuation on convex bodies is very classical; valuations on a class of functions have been introduced and studied by M. Ludwig and others. We study an explicit relation between continuous valuations on convex functions which are invariant under adding arbitrary linear functionals, and translation invariant continuous valuations on convex bodies. More precisely, we construct a natural linear map from the former space to the latter and prove that it has dense image and infinite-dimensional kernel. The proof uses the author’s irreducibility theorem and properties of the real Monge–Ampère operators due to A.D. Alexandrov and Z. Blocki. Furthermore we show how to use complex, quaternionic, and octonionic Monge–Ampère operators to construct more examples of continuous valuations on convex functions in an analogous way.

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.

On the frame of the unit ball of Banach spaces

2014 ◽  
Vol 12 (11) ◽  
Author(s):  
Ryotaro Tanaka
Keyword(s):  
Unit Ball ◽  
Banach Spaces ◽  
Calculation Method ◽  
Extreme Points ◽  
Two Dimensional ◽  
Infinite Dimensional

AbstractThe notion of the frame of the unit ball of Banach spaces was introduced to construct a new calculation method for the Dunkl-Williams constant. In this paper, we characterize the frame of the unit ball by using k-extreme points and extreme points of the unit ball of two-dimensional subspaces. Furthermore, we show that the frame of the unit ball is always closed, and is connected if the dimension of the space is not less than three. As infinite dimensional examples, the frame of the unit balls of c 0 and ℓ p are determined.

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 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
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