Sharp Hessian comparison constant for convex four-body central configurations

Prove that the largest eigenvalue of the matrix G is strictly less than one third at every convex planar four-body central configuration, thereby establishing the uniform bound Q≥(2/3)K and proving that the constant 2/3 is optimal.

Background

The paper proves a uniform lower bound Q≥(13/32)K, where Q is the Hessian quadratic form at a convex four-body central configuration and K is its nonnegative radial part. The authors show that the optimal universal constant is at most 2/3 by analyzing a family in which two masses tend to zero and the smallest generalized eigenvalue tends to 2/3.

The matrix G controls the generalized eigenvalues of (Q,K): if μ1 is its largest eigenvalue, then the smallest generalized eigenvalue is 1−μ1. Numerical evidence suggests that μ1<1/3 throughout the family of convex configurations, with the supremum 1/3 approached only at a boundary corner. Establishing this inequality analytically would improve the proven constant and identify the optimal value.

References

(G)<\frac13 at every convex CC. Consequently Q\ge\frac23K at every convex CC, and by Proposition~\ref{prop:corner}(d) the constant \frac23 is optimal. The following floating-point observations support the conjecture; they are not part of any proof.

— Uniqueness of four-body convex central configurations for all masses  (2609.35632 - Tomar, 28 Sep 2026) in Conjecture 1, Section 7.2, “The corner and the optimal constant”

A proof without computer assistance. By Proposition~\ref{prop:majorant}(c) and the argument of Section~\ref{sec:proofs}, uniqueness follows as soon as $1$ is not an eigenvalue of $G$ at any convex CC, while numerically $(G)<\frac13$. The margin is large, but we do not know a conceptual reason for it.

— Uniqueness of four-body convex central configurations for all masses  (2609.35632 - Tomar, 28 Sep 2026) in Open questions, item (2), Section 7.2, “Open questions”

Is there an analogue of Theorem~B for some class of Dziobek CCs?

— Uniqueness of four-body convex central configurations for all masses  (2609.35632 - Tomar, 28 Sep 2026) in Open questions, item (3), Section 7.2, “Open questions”