Equilateral sets in infinite dimensional Banach spaces

Mercourakis, Sophocles; Vassiliadis, G.

doi:10.1090/s0002-9939-2013-11746-6

Cited by 20 publications

(62 citation statements)

References 17 publications

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“…Those results are the non-separable counterparts to [26, Section 5.1], with essentially the same proofs. Let us also mention [52,Theorem 3], where similar renorming techniques are shown to produce infinite equilateral sets.…”

Section: Combinatorial Analysismentioning

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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Section: Combinatorial Analysismentioning

confidence: 99%

Separated sets and Auerbach systems in Banach spaces

Hájek

Kania

Russo

2020

Trans. Amer. Math. Soc.

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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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“…Equilateral sets have been studied mainly in finite dimensional spaces, see [13], [15] and [14] for a survey on equilateral sets. More recently there are also results on infinite dimensions, see [11], [7] and also on maximal equilateral sets, see [16].…”

Section: Introductionmentioning

confidence: 98%

“…Remarks 2 (1) As was shown by Todorcevic assuming Martin's Axiom and the negation of the continuum hypothesis, if K is compact and non metrizable then the space C(K) admits an uncountable (bounded) biorthogonal system ( [17], Th.11). So by using Theorem 3 of [11], the space C(K) can be given an equivalent norm that admits an uncountable equilateral set.…”

mentioning

confidence: 99%

Equilateral sets in Banach spaces of the form $C(K)$

Mercourakis

Vassiliadis

2016

Studia Math.

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We show that for "most" compact non metrizable spaces, the unit ball of the Banach space C(K) contains an uncountable 2-equilateral set. We also give examples of compact non metrizable spaces K such that the minimum cardinality of a maximal equilateral set in C(K) is countable. γ∈Γ K γ of nontrivial compact metric 2010 Mathematics Subject Classification: Primary 46B20, 46E15;Secondary 46B26,54D30. Key words and phrases: equilateral set, maximal equilateral set, linked family.2. Let K be a compact totally disconnected space, then the family {(V, K \V ) :V is a clopen subset of K} is linked.

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Equilateral sets in infinite dimensional Banach spaces

Abstract: We show that every Banach space X containing an isomorphic copy of c 0 has an infinite equilateral set and also that if X has a bounded biorthogonal system of size α then it can be renormed so as to admit an equilateral set of equal size.

Cited by 20 publications

References 17 publications

Separated sets and Auerbach systems in Banach spaces

Separated sets and Auerbach systems in Banach spaces

Combinatorial Distance Geometry in Normed Spaces

Equilateral sets in Banach spaces of the form $C(K)$

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