In this paper, we provide a rigorous computer-assisted proof (CAP) of the conjecture
that in the planar four-body problem 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 (IFT). 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 neighborhood.
Keywords:
central configuration, convex central configuration, uniqueness, $N$-body problem, Krawczyk operator, implicit function theorem
Citation:
Sun S., Xie Z., You P., On the Uniqueness of Convex Central Configurations in the Planar $4$-Body Problem, Regular and Chaotic Dynamics,
2023, Volume 28, Numbers 4-5,
pp. 512-532