Uncountable equilateral sets in Banach spaces of the form C(K)

Koszmider, Piotr

doi:10.1007/s11856-018-1637-9

Cited by 11 publications

(49 citation statements)

References 28 publications

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“…There are other results that cast some doubt on Conjecture 3: the existence of small maximal equilateral sets (Section 3.2) and the existence of infinite-dimensional normed spaces that do not have infinite equilateral sets, first shown by Terenzi [198,199]; see also Glakousakis and Mercourakis [79]. (For more on equilateral sets in infinite-dimensional space, see [71,107,131,132]. )…”

Section: Equilateral Setsmentioning

confidence: 99%

Combinatorial Distance Geometry in Normed Spaces

Swanepoel

2018

Bolyai Society Mathematical Studies

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We survey problems and results from combinatorial geometry in normed spaces, concentrating on problems that involve distances. These include various properties of unit-distance graphs, minimum-distance graphs, diameter graphs, as well as minimum spanning trees and Steiner minimum trees. In particular, we discuss translative kissing (or Hadwiger) numbers, equilateral sets, and the Borsuk problem in normed spaces. We show how to use the angular measure of Peter Brass to prove various statements about Hadwiger and blocking numbers of convex bodies in the plane, including some new results. We also include some new results on thin cones and their application to distinct distances and other combinatorial problems for normed spaces.

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Section: Equilateral Setsmentioning

confidence: 99%

Combinatorial Distance Geometry in Normed Spaces

Swanepoel

2018

Bolyai Society Mathematical Studies

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“…Nyikos ([29, Example 6.17]) constructed, assuming Jensen's Diamond Principle ♦, a non-metrisable, compact manifold K whose each non-metrisable, closed subspace contains a copy of the unit interval (thus, it is not totally disconnected). Therefore, one cannot hope to prove such a theorem about totally disconnected subspaces in ZFC only (this also follows from the main result of [19]). However Nyikos' manifold is non-separable, hence by Proposition 4.3, the number eo(C(K)) is uncountable.…”

Section: Spaces Of Continuous Functionsmentioning

confidence: 99%

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

Kania

Kochanek

2016

Studia Math.

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Let $X$ be a Banach space. We study the circumstances under which there exists an uncountable set $\mathcal A\subset X$ of unit vectors such that $\|x-y\|>1$ for distinct $x,y\in \mathcal A$. 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 $\|x-y\|\geqslant 1+\varepsilon$ for some $\varepsilon>0$ 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.Comment: to appear in Studia Mat

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“…Koszmider proved that an 'uncountable' version of the Elton-Odell theorem for nonseparable C(K)-spaces is independent of ZFC [44]. Interestingly, if the unit sphere of a C(K)-space contains a (1 + ε)-separated subset for some ε > 0, then it also contains a 2-separated subset of the same cardinality ([53, Theorem 1]).…”

Section: Introductionmentioning

confidence: 99%

Separated sets and Auerbach systems in Banach spaces

Hájek

Kania

Russo

2020

Trans. Amer. Math. Soc.

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The paper elucidates the relationship between the density of a Banach space and possible sizes of Auerbach systems and well-separated subsets of its unit sphere. For example, it is proved that for a large enough space X, the unit sphere S X always contains an uncountable (1+)-separated subset. In order to achieve this, new results concerning the existence of large Auerbach systems are established, that happen to be sharp for the class of WLD spaces. In fact, we offer the first consistent example of a non-separable WLD Banach space that contains no uncountable Auerbach system, as witnessed by a renorming of c 0 (ω 1 ). Moreover, the following optimal results for the classes of, respectively, reflexive and super-reflexive spaces are established: the unit sphere of an infinite-dimensional reflexive space contains a symmetrically (1 + ε)-separated subset of any regular cardinality not exceeding the density of X; should the space X be superreflexive, the unit sphere of X contains such a subset of cardinality equal to the density of X. The said problem is studied for other classes of spaces too, including WLD spaces, RNP spaces, or strictly convex ones.

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Uncountable equilateral sets in Banach spaces of the form C(K)

Cited by 11 publications

References 28 publications

Combinatorial Distance Geometry in Normed Spaces

Combinatorial Distance Geometry in Normed Spaces

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

Separated sets and Auerbach systems in Banach spaces

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