The negative energy N-body problem has finite diameter

The Jacobi-Maupertuis metric provides a reformulation of the classical N-body problem as a geodesic flow on an energy-dependent metric space denoted $M_E$ where $E$ is the energy of the problem. We show that $M_E$ has finite diameter for $E < 0$. Consequently $M_E$ has no metric rays. Motivation comes from work of Burgos- Maderna and Polimeni-Terracini for the case $E \ge 0$ and from a need to correct an error made in a previous ``proof''. We show that $M_E$ has finite diameter for $E < 0$ by showing that there is a constant $D$ such that all points of the Hill region lie a distance $D$ from the Hill boundary. (When $E \ge 0$ the Hill boundary is empty.) The proof relies on a game of escape which allows us to quantify the escape rate from a closed subset of configuration space, and the reduction of this game to one of escaping the boundary of a polyhedral convex cone into its interior.