John Ellipsoid and the Center of Mass of a Convex Body

TL;DR

This paper demonstrates that the center of mass of certain convex bodies and polytopes can lie outside their John ellipsoid when inflated by specific factors, challenging assumptions used in convex body algorithms.

Contribution

It provides the first known examples where the center of mass is outside the inflated John ellipsoid, revealing limitations in geometric assumptions for convex body algorithms.

Findings

Existence of convex bodies with center of mass outside inflated John ellipsoid

Construction of polytopes with many facets where center of mass is outside inflated John ellipsoid

Quantitative bounds on inflation factors for these counterexamples

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 $(1-C\sqrt{\frac{\log(n)} {n}})n$ 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 $O(\frac{n}{\log(n)})$ times about the center of $B_P$.