The negative energy N-body problem has finite diameter

The Jacobi-Maupertuis metric lets us reformulate the classical N-body problem at fixed energy E as a geodesic flow problem on a space whose metric depends parametrically on E. We only consider the case E < 0 in which case there is a non-empty Hill boundary along which the metric degenerates. Our main result is the resulting metric space has finite diameter. As a corollary the space admits no metric rays, answering a question of Burgos (Proc Amer Math Soc 150: 1729-1733, 2022). This main result is an immediate corollary of a theorem asserting that all points of the space are a fixed bounded distance from the Hill boundary. Our proof of this last theorem relies ultimately on a game of escape from the boundary of a polyhedral convex cone in a Euclidean space into the interior of said cone. Motivation for our work comes from that of Maderna (Ann Math 192: 499-550, 2020) and from the desire to right a wrong promulgated in Montgomery (Regul Chaot Dyn 28: 374-394, 2023).