Central configurations play an important role in the dynamics of the $n$-body problem: they occur as relative equilibria and as asymptotic configurations in colliding trajectories. We illustrate how they can be found as projective fixed points of self-maps defined on the shape space, and some results on the inverse problem in dimension $1$, i.e. finding (positive or real) masses which make a given collinear configuration central. This survey article introduces readers to the recent results of the author, also unpublished, showing an application of the fixed point theory.
"Fixed points and the inverse problem for central configurations." Topol. Methods Nonlinear Anal. 56 (2) 579 - 588, 2020. https://doi.org/10.12775/TMNA.2020.034