Sailing over three problems of Koszmider

Sánchez, Félix Cabello; Castillo, Jesús M. F.; Marciszewski, Witold; Plebanek, Grzegorz; Salguero-Alarcón, Alberto

doi:10.1016/j.jfa.2020.108571

Cited by 5 publications

(2 citation statements)

References 21 publications

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“…Finally, let us comment on the isomorphic theory structure of our spaces (For the Kottman constant in the isomorphic context, see [4]). Using the arguments of Section 5 of [2] (cf. [35]) one can see that there are pairwise nonisomorphic spaces satisfying Theorem 1.…”

Section: Introductionmentioning

confidence: 99%

“… in the norm admits a -equilateral set of cardinality continuum which is an Auerbach system (just characteristic functions of the elements of the almost disjoint family ). Although, as mentioned above, there are many nonisomorphic Banach spaces of the form for an almost disjoint families , under Martin’s axiom and the negation of all spaces are pairwise isomorphic for of cardinality , in particular can be isomorphic to with being -embeddable and being Luzin ([2]). This, for example, provides equivalent renormings of for any Luzin family which satisfy Theorem 1.…”

Section: Introductionmentioning

confidence: 99%

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Banach Spaces in Which Large Subsets of Spheres Concentrate

Koszmider

2022

J. Inst. Math. Jussieu

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We construct a nonseparable Banach space $\mathcal {X}$ (actually, of density continuum) such that any uncountable subset $\mathcal {Y}$ of the unit sphere of $\mathcal {X}$ contains uncountably many points distant by less than $1$ (in fact, by less then $1-\varepsilon $ for some $\varepsilon>0$ ). This solves in the negative the central problem of the search for a nonseparable version of Kottman’s theorem which so far has produced many deep positive results for special classes of Banach spaces and has related the global properties of the spaces to the distances between points of uncountable subsets of the unit sphere. The property of our space is strong enough to imply that it contains neither an uncountable Auerbach system nor an uncountable equilateral set. The space is a strictly convex renorming of the Johnson–Lindenstrauss space induced by an $\mathbb {R}$ -embeddable almost disjoint family of subsets of $\mathbb {N}$ . We also show that this special feature of the almost disjoint family is essential to obtain the above properties.

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