Banach spaces in which large subsets of spheres concentrate

Abstract

We construct a nonseparable Banach space X (actually, of density continuum) such that any uncountable subset Y of the unit sphere of X contains uncountably many points distant by less than 1 (in fact, by less then 1 − ε for some ε > 0). This solves in the negative the central problem of the search for a nonseparable version of Kottman's theorem which so far produced many deep positive results for special classes of Banach spaces and related the global properties of the spaces to the distances between pairs of points of uncountable subsets of the unit sphere. The property of our space is strong enough to imply that it contains no uncountable Auerbach system nor an uncountable equilateral set. The space is a strictly convex renorming of the Johnson-Lindenstrauss space induced by an R-embeddable almost disjoint family of subsets of N. We also show that this special property of the almost disjoint family is essential to obtain the above properties.

Large Banach spaces with no infinite equilateral sets

“…space which contains an isomorphic copy of 𝑐 0 admits an infinite equilateral set, we conclude that spaces of [12] admit such infinite sets.…”

“…Suppose {𝛿 𝑛 ∶ 𝑛 ∈ ℕ} ∪ {𝛿 𝜔 } forms such a set, where 𝛿 𝜉 < 𝛿 𝜂 if 𝜉 < 𝜂 for all 𝜉, 𝜂 < 𝜔 + 1. Since the supports of 𝑥 𝛼 𝛿 ,𝛿 s for 𝛿 ∈ 𝐶 ′′ are pairwise disjoint by (12) For distinct 𝑡, 𝑡 ′ ∈ [0, 1] the intersection of supports of 𝑓 𝑡 and 𝑓 𝑡 ′ is included in {{𝑡, 𝑡 ′ }}. For 𝑡 < 𝑡 ′ the value of 𝑓 𝑡 − 𝑓 𝑡 ′ at {𝑡, 𝑡 ′ } is 1 − (−1) = 2 if 𝑐({𝑡, 𝑡 ′ }) = 1 and it is 0 otherwise, so…”

“…So for 𝛿 ∈ 𝐶 we can find 𝛼 𝛿 < 𝜅 such that 𝑥 𝛼 𝛿 ,𝛿 ≠ 0 and moreover we may make sure that 𝛼 𝛿 ≠ 𝛼 𝛿 ′ for any 𝛿 < 𝛿 ′ in 𝐶. Next we find 𝐶 ′ ⊆ 𝐶 of cardinality 𝜅 such that (12) 𝑠𝑢𝑝𝑝(𝑥 𝛼 𝛿 ′ ,𝛿 ′ ) ⊆ {𝑡 𝜉 ∶ 𝛿 ′ ⩽ 𝜉 < 𝛿} for any 𝛿 ′ < 𝛿 and 𝛿, 𝛿 ′ ∈ 𝐶 ′ . This can be done by recursion taking at the inductive step the next 𝛿 ∈ 𝐶 such that the supports of the previous 𝑥 𝛼 𝛿 ′ ,𝛿 ′ s are included in {𝑡 𝜉 ∶ 𝜉 < 𝛿}.…”

“…In particular, this solves the question of whether there is a nonseparable Banach space with no uncountable equilateral set ( [11,14], Problem 293 of [7]). Another absolute construction of a nonseparable Banach space with no uncountable equilateral set is being presented at the same time in a paper by the first author [12]. However, that is a renorming of a space 𝐶 0 (𝐾) for 𝐾 locally compact and scattered, so it is 𝑐 0 -saturated (by [15]).…”

“…there is 𝑛 ∈ ℕ such that ‖𝑥 𝛼 𝛿 𝜔 |𝑠𝑢𝑝𝑝(𝑥 𝛼 𝛿 𝑛 ,𝛿 𝑛 )‖ 1 ⩽ 𝜀∕2.Also by(13) we have 𝑡 𝜃 𝛿 𝜔 ∉ 𝑠𝑢𝑝𝑝(𝑥 𝛼 𝛿 𝑛 ,𝛿 𝑛 ) ∪ 𝑠𝑢𝑝𝑝(𝑥 𝛼 𝛿 𝜔 ,𝛿 𝜔 ), where the union is disjoint by(12). So by Claim 9 we have‖𝑥 𝛼 𝛿 𝜔 − 𝑥 𝛼 𝛿 𝑛 ‖ 𝑇 > ‖𝑥 𝛼 𝛿 𝜔 − 𝑥 𝛼 𝛿 𝑛 ‖ 1 ⩾ ‖𝑥 𝛼 𝛿 𝜔 ,𝛿 𝜔 ‖ 1 + |𝑥 𝛼 𝛿 𝜔 (𝑡 𝜃 𝛿 𝜔 )| + ‖𝑥 𝛼 𝛿 𝑛 ,𝛿 𝑛 ‖ 1 − 𝜀∕2 ⩾ 𝑟which contradicts the choice of {𝛿 𝑛 ∶ 𝑛 ∈ ℕ} ∪ {𝛿 𝜔 } as 0-monochromatic.…”