On the Löwner-John Ellipsoids of the Metric Polytope

The collection of all n-point metric spaces of diameter \(\le 1\) $\le 1$ constitutes a polytope \(\mathcal {M}_n \subset \mathbb {R}^{\left( {\begin{array}{c}n\\ 2\end{array}}\right) }\) ${\mathcal{M}}_n\subset {\mathbb{R}}^{\left(\begin{array}{c}n\\ 2\end{array}\right)}$ , called the Metric Polytope. In this paper, we consider the best approximations of \(\mathcal {M}_n\) ${\mathcal{M}}_n$ by ellipsoids. We give an exact explicit description of the largest volume ellipsoid contained in \(\mathcal {M}_n\) ${\mathcal{M}}_n$ . When inflated by a factor of \(\Theta (n)\) $\mathrm{\Theta }(n)$ , this ellipsoid contains \(\mathcal {M}_n\) ${\mathcal{M}}_n$ . It also turns out that the least volume ellipsoid containing \(\mathcal {M}_n\) ${\mathcal{M}}_n$ is a ball. When shrunk by a factor of \(\Theta (n)\) $\mathrm{\Theta }(n)$ , the resulting ball is contained in \(\mathcal {M}_n\) ${\mathcal{M}}_n$ . We note that the general theorems on such ellipsoid posit only that the pertinent inflation/shrinkage factors can be made as small as \(O(n^2)\) $O(n^2)$ .