scholarly journals Support theorems in abstract settings

2018 ◽  
Vol 93 (1-2) ◽  
pp. 215-240
Author(s):  
Andrzej Olbrys ◽  
Zsolt Pales
Keyword(s):  
Support Theorems

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.

Composition of Skeletons and Support Theorems

1997 ◽  
pp. 21-33
Author(s):  
María Emilia Caballero ◽  
Begoña Fernández ◽  
David Nualart
Keyword(s):  
Support Theorems
Sign in / Sign up

Export Citation Format

Share Document