Uniqueness of four-body convex central configurations for all masses
Abstract
We prove that for every choice of four positive masses and every cyclic ordering of the bodies there is exactly one strictly convex planar central configuration with that ordering, up to similarity. This proves a conjecture of Albouy, Fu and Sun, also called the Simó-Yoccoz conjecture, and answers Problem 10 in the list of Albouy, Cabral and Santos. Uniqueness follows by a covering argument from the nondegeneracy of these configurations. We prove nondegeneracy by hand: Dziobek's relations reduce it to an eigenvalue bound for a 2×2 matrix that depends only on the shape of the configuration. As consequences, every degenerate planar four-body central configuration is concave, and the known symmetry theorems for kites, isosceles trapezoids and rhombi follow in a few lines.
Related
- Planar Central Configurations of Four Equal Masses: Classification and Morse Indices
- Uniqueness of convex central configurations in the planar (1 + 4)-body problem
- Finiteness of planar central configurations of five bodies with positive masses
- Finiteness of planar central configurations of six bodies for all positive masses: Part I, exceptional-mass first limits
- Finiteness of planar central configurations of five bodies for all positive masses: On Smale's sixth problem
BibTeX
@misc{tomar2026uniqueness,
title = {{Uniqueness of four-body convex central configurations for all masses}},
author = {Tejasvi Singh Tomar},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2609.35632},
eprint = {2609.35632},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.35632},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Uniqueness of four-body convex central configurations for all masses
AU - Tejasvi Singh Tomar
PY - 2026
DA - 2026/10/29
PB - arXiv
DO - 10.48550/arxiv.2609.35632
UR - https://arxiv.org/abs/2609.35632
AB - We prove that for every choice of four positive masses and every cyclic ordering of the bodies there is exactly one strictly convex planar central configuration with that ordering, up to similarity. This proves a conjecture of Albouy, Fu and Sun, also called the Simó-Yoccoz conjecture, and answers Problem 10 in the list of Albouy, Cabral and Santos. Uniqueness follows by a covering argument from the nondegeneracy of these configurations. We prove nondegeneracy by hand: Dziobek's relations reduce it to an eigenvalue bound for a 2×2 matrix that depends only on the shape of the configuration. As consequences, every degenerate planar four-body central configuration is concave, and the known symmetry theorems for kites, isosceles trapezoids and rhombi follow in a few lines.
ER -