On the Löwner-John Ellipsoids of the Metric Polytope

The collection of all n-point metric spaces of diameter ≤1 constitutes a polytope M_n⊂Rⁿ², called the Metric Polytope. In this paper, we consider the best approximations of M_n by ellipsoids. We give an exact explicit description of the largest volume ellipsoid contained in M_n. When inflated by a factor of Θ(n), this ellipsoid contains M_n. It also turns out that the least volume ellipsoid containing M_n is a ball. When shrunk by a factor of Θ(n), the resulting ball is contained in 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(n²).