A direct proof that ${\ell }_{\infty }^{(3)}$ has generalized roundness zero

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 $0<p\le 2$. Lennard, Tonge and Weston gave an indirect proof that ${\ell }_{\infty }^{(3)}$ 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 ${\ell }_{\infty }^{(3)}$ has generalized roundness zero. This provides insight into the combinatorial geometry of ${\ell }_{\infty }^{(3)}$ 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.

The first version of this paper had the title "The generalized roundness of ${\ell }_{\infty }^{(3)}$ revisited". This version includes some minor modifications of the text and corrections to several typographic errors