A direct proof that ${\mathrm{\ell }}_{\mathrm{\infty }}^{(3)}$ 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 $0 < p \leq 2$\mathrm{.}Lennard,TongeandWestongaveanindirectproofthat$\ell_{\infty}^{(3)}$ has generalized roundness zero by appealing to highly non-trivial isometric embedding theorems of Bretagnolle Dacunha-Castelle and Krivine, and Misiewicz. In this paper we give a direct proof that ${\mathrm{\ell }}_{\mathrm{\infty }}^{(3)}$ has generalized roundness zero. This provides insight into the combinatorial geometry of ${\mathrm{\ell }}_{\mathrm{\infty }}^{(3)}$ 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.