A direct proof that (3) has generalized roundness zero

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.