scholarly journals Removable sets of support points of convex sets in Banach spaces

1987 ◽  
Vol 99 (2) ◽  
pp. 319-319 ◽  
Author(s):  
R. R. Phelps
Keyword(s):  
Banach Spaces ◽  
Convex Sets ◽  
Support Points ◽  
Removable Sets

Counterexamples Concerning Support Theorems for Convex Sets in Hilbert Space

1988 ◽  
Vol 31 (1) ◽  
pp. 121-128 ◽  
Author(s):  
R. R. Phelps
Keyword(s):  
Banach Space ◽  
Hilbert Space ◽  
Convex Function ◽  
Convex Subset ◽  
Convex Sets ◽  
Lower Semicontinuous ◽  
Nonempty Closed Convex Subset ◽  
Common Point ◽  
Support Points ◽  
Support Theorems

AbstractThe Bishop-Phelps theorem guarantees the existence of support points and support functionals for a nonempty closed convex subset of a Banach space; equivalently, it guarantees the existence of subdifferentials and points of subdifferentiability of a proper lower semicontinuous convex function on a Banach space. In this note we show that most of these results cannot be extended to pairs of convex sets or functions, even in Hilbert space. For instance, two proper lower semicontinuous convex functions need not have a common point of subdifferentiability nor need they have a subdifferential in common. Negative answers are also obtained to certain questions concerning density of support points for the closed sum of two convex subsets of Hilbert space.

scholarly journals A Characterization of Polynomially Convex Sets in Banach Spaces

2017 ◽  
Vol 72 (4) ◽  
pp. 2013-2021
Author(s):  
Mortaza Abtahi ◽  
Sara Farhangi
Keyword(s):  
Banach Spaces ◽  
Convex Sets ◽  
Polynomially Convex

(I)-envelopes of closed convex sets in Banach spaces

2007 ◽  
Vol 162 (1) ◽  
pp. 157-181 ◽  
Author(s):  
Ondřej F. K. Kalenda
Keyword(s):  
Banach Spaces ◽  
Convex Sets

scholarly journals Poincaré-Hopf type formulas on convex sets of Banach spaces

2009 ◽  
Vol 34 (2) ◽  
pp. 213 ◽  
Author(s):  
Thomas Bartsch ◽  
E. Norman Dancer
Keyword(s):  
Banach Spaces ◽  
Convex Sets

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.

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