John Ellipsoid and the Center of Mass of a Convex Body

Abstract

It is natural to ask whether the center of mass of a convex body \(K\subset {\mathbb {R}}^n\) lies in its John ellipsoid \(B_K\), i.e., in the maximal volume ellipsoid contained in K. This question is relevant to the efficiency of many algorithms for convex bodies. In this paper, we obtain an unexpected negative result. There exists a convex body \(K\subset {\mathbb {R}}^n\) such that its center of mass does not lie in the John ellipsoid \(B_K\) inflated \(\bigl (1-C\sqrt{\frac{\log (n)}{n}}\bigr )n\) times about the center of \(B_K\). Moreover, there exists a polytope \(P \subset {\mathbb {R}}^n\) with \(O(n^2)\) facets whose center of mass is not contained in the John ellipsoid \(B_P\) inflated \(O\bigl (\frac{n}{\log (n)}\bigr )\) times about the center of \(B_P\).