Distances to convex sets

Antonio S. Granero, and Marcos Sánchez. "Distances to convex sets." Studia Mathematica 182.2 (2007): 165-181. <http://eudml.org/doc/286267>.

@article{AntonioS2007,
abstract = {If X is a Banach space and C a convex subset of X*, we investigate whether the distance $d̂(\overline\{co\}^\{w*\}(K),C):= sup\{inf\{||k-c||: c ∈ C\}: k ∈ \overline\{co\}^\{w*\}(K)\}$ from $\overline\{co\}^\{w*\}(K)$ to C is M-controlled by the distance d̂(K,C) (that is, if $d̂(\overline\{co\}^\{w*\}(K),C) ≤ M d̂(K,C)$ for some 1 ≤ M < ∞), when K is any weak*-compact subset of X*. We prove, for example, that: (i) C has 3-control if C contains no copy of the basis of ℓ₁(c); (ii) C has 1-control when C ⊂ Y ⊂ X* and Y is a subspace with weak*-angelic closed dual unit ball B(Y*); (iii) if C is a convex subset of X and X is considered canonically embedded into its bidual X**, then C has 5-control inside X**, in general, and 2-control when K ∩ C is weak*-dense in C.},
author = {Antonio S. Granero, Marcos Sánchez},
journal = {Studia Mathematica},
keywords = {convex sets; distances; Krein-Shmul'yan theorem},
language = {eng},
number = {2},
pages = {165-181},
title = {Distances to convex sets},
url = {http://eudml.org/doc/286267},
volume = {182},
year = {2007},
}

TY - JOUR
AU - Antonio S. Granero
AU - Marcos Sánchez
TI - Distances to convex sets
JO - Studia Mathematica
PY - 2007
VL - 182
IS - 2
SP - 165
EP - 181
AB - If X is a Banach space and C a convex subset of X*, we investigate whether the distance $d̂(\overline{co}^{w*}(K),C):= sup{inf{||k-c||: c ∈ C}: k ∈ \overline{co}^{w*}(K)}$ from $\overline{co}^{w*}(K)$ to C is M-controlled by the distance d̂(K,C) (that is, if $d̂(\overline{co}^{w*}(K),C) ≤ M d̂(K,C)$ for some 1 ≤ M < ∞), when K is any weak*-compact subset of X*. We prove, for example, that: (i) C has 3-control if C contains no copy of the basis of ℓ₁(c); (ii) C has 1-control when C ⊂ Y ⊂ X* and Y is a subspace with weak*-angelic closed dual unit ball B(Y*); (iii) if C is a convex subset of X and X is considered canonically embedded into its bidual X**, then C has 5-control inside X**, in general, and 2-control when K ∩ C is weak*-dense in C.
LA - eng
KW - convex sets; distances; Krein-Shmul'yan theorem
UR - http://eudml.org/doc/286267
ER -