Doust, I., Sanchez, S. & Weston, A. (2015). A direct proof that (3) has generalized roundness zero. Expositiones Mathematicae,33(2), 259-267. Retrieved from https://doi.org/10.1016/j.exmath.2014.06.001
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.
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