# On the Uniqueness of Convex Central Configurations in the Planar $4$-Body Problem

## TL;DR

This paper rigorously proves the uniqueness of convex central configurations in the planar 4-body problem for fixed masses using computer-assisted methods, combining the Krawczyk operator and the implicit function theorem.

## Contribution

It introduces a novel combination of interval analysis with the implicit function theorem to establish uniqueness in a parameter space for the 4-body problem.

## Key findings

- Confirmed the uniqueness of convex central configurations for fixed masses
- Developed a method combining interval analysis with the implicit function theorem
- Extended the results to neighborhoods around specific mass points

## Abstract

In this paper, we provide a rigorous computer-assisted proof (CAP) of the conjecture that there exists a unique convex central configuration for any four fixed positive masses in a given order belonging to a closed domain in the mass space. The proof employs the Krawczyk operator and the implicit function theorem. Notably, we demonstrate that the implicit function theorem can be combined with interval analysis, enabling us to estimate the size of the region where the implicit function exists and extend our findings from one mass point to its surrounding neighborhood.