Uniqueness of four-body convex central configurations for all masses
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 answers Problem 10 in the list of Albouy, Cabral and Santos, which Santoprete calls the Simó--Yoccoz conjecture. The main step is a uniform lower bound for the Hessian at convex central configurations: it is at least one quarter of its radial part, which vanishes only on translations and rotations. Dziobek's relations turn the indefinite part of the Hessian into a negative multiple of a…
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