Distance Estimates for Action-Minimizing Solutions of the $N$-Body Problem

In this paper we provide estimates for mutual distances of periodic solutions for the Newtonian $N$-body problem. Our estimates are based on masses, total variations of turning angles for relative positions, and predetermined upper bounds for action values. Explicit formulae will be proved by iterative arguments. We demonstrate some applications to action-minimizing solutions for three- and four-body problems.