Counterexamples Concerning Support Theorems for Convex Sets in Hilbert Space
Canadian mathematical bulletin, Tome 31 (1988) no. 1, pp. 121-128
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The 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.
Phelps, R. R. Counterexamples Concerning Support Theorems for Convex Sets in Hilbert Space. Canadian mathematical bulletin, Tome 31 (1988) no. 1, pp. 121-128. doi: 10.4153/CMB-1988-019-1
@article{10_4153_CMB_1988_019_1,
author = {Phelps, R. R.},
title = {Counterexamples {Concerning} {Support} {Theorems} for {Convex} {Sets} in {Hilbert} {Space}},
journal = {Canadian mathematical bulletin},
pages = {121--128},
year = {1988},
volume = {31},
number = {1},
doi = {10.4153/CMB-1988-019-1},
url = {http://geodesic.mathdoc.fr/articles/10.4153/CMB-1988-019-1/}
}
TY - JOUR AU - Phelps, R. R. TI - Counterexamples Concerning Support Theorems for Convex Sets in Hilbert Space JO - Canadian mathematical bulletin PY - 1988 SP - 121 EP - 128 VL - 31 IS - 1 UR - http://geodesic.mathdoc.fr/articles/10.4153/CMB-1988-019-1/ DO - 10.4153/CMB-1988-019-1 ID - 10_4153_CMB_1988_019_1 ER -
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