# Classifying Four-Body Convex Central Configurations

## TL;DR

This paper provides a comprehensive classification of convex central configurations in the Newtonian four-body problem, identifying their geometric properties and establishing a three-dimensional structure for configurations with positive masses.

## Contribution

It introduces a detailed classification of convex four-body central configurations, including symmetry and geometric properties, and proves the three-dimensional nature of their configuration set.

## Key findings

- Convex four-body central configurations form a three-dimensional set.
- Configurations with positive masses can be described as a graph over a union of elementary regions.
- Specific configurations like kite, trapezoidal, and co-circular are characterized and classified.

## Abstract

We classify the full set of convex central configurations in the Newtonian four-body problem. Particular attention is given to configurations possessing some type of symmetry or defining geometric property. Special cases considered include kite, trapezoidal, co-circular, equidiagonal, orthodiagonal, and bisecting-diagonal configurations. Good coordinates for describing the set are established. We use them to prove that the set of four-body convex central configurations with positive masses is three-dimensional, a graph over a domain $D$ that is the union of elementary regions in $\mathbb{R}^{+^3}$.