A direct proof that (3) has generalized roundness zero

Journal article


Doust, Ian, Sanchez, Stephen and Weston, Anthony. (2015) A direct proof that (3) has generalized roundness zero. Expositiones Mathematicae. 33(2), pp. 259 - 267. https://doi.org/10.1016/j.exmath.2014.06.001
AuthorsDoust, Ian, Sanchez, Stephen and Weston, Anthony
Abstract

Metric spaces of generalized roundness zero have interesting non-embedding properties. For instance, we note that no metric space of generalized roundness zero is isometric to any metric subspace of any L p-space for which 0 < p ≤ 2. Lennard, Tonge and Weston gave an indirect proof that ℓ (3) ∞ has generalized roundness zero by appealing to non-trivial isometric embedding theorems of Bretagnolle, Dacunha-Castelle and Krivine, and Misiewicz. In this paper we give a direct proof that ℓ (3) ∞ has generalized roundness zero. This provides insight into the combinatorial geometry of ℓ (3) ∞ that causes the generalized roundness inequalities to fail. We complete the paper by noting a characterization of real quasi-normed spaces of generalized roundness zero.

Year2015
JournalExpositiones Mathematicae
Journal citation33 (2), pp. 259 - 267
ISSN0723-0869
Digital Object Identifier (DOI)https://doi.org/10.1016/j.exmath.2014.06.001
Page range259 - 267
Research GroupSchool of Education
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