On the Löwner-John Ellipsoids of the Metric Polytope

Abstract

AbstractThe collection of all n-point metric spaces of diameter $$\le 1$$ ≤ 1 constitutes a polytope $$\mathcal {M}_n \subset \mathbb {R}^{\left( {\begin{array}{c}n\\ 2\end{array}}\right) }$$ M n ⊂ R n 2 , called the Metric Polytope. In this paper, we consider the best approximations of $$\mathcal {M}_n$$ M n by ellipsoids. We give an exact explicit description of the largest volume ellipsoid contained in $$\mathcal {M}_n$$ M n . When inflated by a factor of $$\Theta (n)$$ Θ ( n ) , this ellipsoid contains $$\mathcal {M}_n$$ M n . It also turns out that the least volume ellipsoid containing $$\mathcal {M}_n$$ M n is a ball. When shrunk by a factor of $$\Theta (n)$$ Θ ( n ) , the resulting ball is contained in $$\mathcal {M}_n$$ 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 ) .