scholarly journals Support theorems in abstract settings

2018 ◽  
Vol 93 (1-2) ◽  
pp. 215-240
Author(s):  
Andrzej Olbrys ◽  
Zsolt Pales
Keyword(s):  
1988 ◽  
Vol 31 (1) ◽  
pp. 121-128 ◽  
Author(s):  
R. R. Phelps

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.


Author(s):  
María Emilia Caballero ◽  
Begoña Fernández ◽  
David Nualart
Keyword(s):  

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