Uncountable sets of unit vectors that are separated by more than 1

Tomasz Kania, and Tomasz Kochanek. "Uncountable sets of unit vectors that are separated by more than 1." Studia Mathematica 232.1 (2016): 19-44. <http://eudml.org/doc/286656>.

@article{TomaszKania2016,
abstract = {Let X be a Banach space. We study the circumstances under which there exists an uncountable set 𝓐 ⊂ X of unit vectors such that ||x-y|| > 1 for any distinct x,y ∈ 𝓐. We prove that such a set exists if X is quasi-reflexive and non-separable; if X is additionally super-reflexive then one can have ||x-y|| ≥ slant 1 + ε for some ε > 0 that depends only on X. If K is a non-metrisable compact, Hausdorff space, then the unit sphere of X = C(K) also contains such a subset; if moreover K is perfectly normal, then one can find such a set with cardinality equal to the density of X; this solves a problem left open by S. K. Mercourakis and G. Vassiliadis (2015).},
author = {Tomasz Kania, Tomasz Kochanek},
journal = {Studia Mathematica},
keywords = {Kottman theorem; Elton-Odell theorem; unit sphere; equilateral set; quasi-reflexive space; super-reflexive space; cardinal function},
language = {eng},
number = {1},
pages = {19-44},
title = {Uncountable sets of unit vectors that are separated by more than 1},
url = {http://eudml.org/doc/286656},
volume = {232},
year = {2016},
}

TY - JOUR
AU - Tomasz Kania
AU - Tomasz Kochanek
TI - Uncountable sets of unit vectors that are separated by more than 1
JO - Studia Mathematica
PY - 2016
VL - 232
IS - 1
SP - 19
EP - 44
AB - Let X be a Banach space. We study the circumstances under which there exists an uncountable set 𝓐 ⊂ X of unit vectors such that ||x-y|| > 1 for any distinct x,y ∈ 𝓐. We prove that such a set exists if X is quasi-reflexive and non-separable; if X is additionally super-reflexive then one can have ||x-y|| ≥ slant 1 + ε for some ε > 0 that depends only on X. If K is a non-metrisable compact, Hausdorff space, then the unit sphere of X = C(K) also contains such a subset; if moreover K is perfectly normal, then one can find such a set with cardinality equal to the density of X; this solves a problem left open by S. K. Mercourakis and G. Vassiliadis (2015).
LA - eng
KW - Kottman theorem; Elton-Odell theorem; unit sphere; equilateral set; quasi-reflexive space; super-reflexive space; cardinal function
UR - http://eudml.org/doc/286656
ER -