Full Fisher–Rao beta-manifold sectional-curvature conjecture

Establish that, for all x,y>0, the sectional curvature K(x,y) of the Fisher–Rao manifold of beta distributions satisfies K(x,y)>-1/2 and is decreasing in both x and y.

Background

The paper considers the Fisher–Rao metric on the beta-distribution parameter space M={(x,y):x,y>0}. Earlier work derived an explicit formula for the sectional curvature K(x,y), proved that it is negative, and established that it is bounded from below. Those results led to the conjecture that the sharp lower bound is −1/2 and that K(x,y) decreases with respect to each parameter.

The paper verifies only the diagonal specialization K(x,x): it proves that this one-variable sectional curvature is strictly decreasing and lies strictly between 0 and −1/2. The full two-variable lower-bound and coordinatewise-monotonicity conjecture therefore remains unresolved in the paper. A later remark reformulates the remaining tasks in terms of an auxiliary function Q(x,y), but those reformulations do not themselves provide a proof.

References

InProposition~5, Theorem~6, Theorem~6, andTheorem~6, the authors established that the sectional curvature $K(x,y)$ is negative and bounded from below, and they subsequently proposed the following conjecture.

For $x,y>0$, the sectional curvature $K(x,y)$ given in~curvature-polygamma-exp satisfies that \begin{enumerate} \item it has a lower bound $-\frac{1}{2}$, accurately, $K(x,y)>-\frac{1}{2}$; \item it is decreasing in both $x$ and $y$. \end{enumerate}

curvature-polygamma-exp:

K(x,y)=14ψ′′(x)ψ′′(y)ψ′′(x+y)[ψ′(x)ψ′′(x)+ψ′(y)ψ′′(y)−ψ′(x+y)ψ′′(x+y)][ψ′(x)ψ′(x+y)+ψ′(y)ψ′(x+y)−ψ′(x)ψ′(y)]2,x,y∈M.K(x,y)=\frac{1}{4}\frac{\psi''(x)\psi''(y)\psi''(x+y) \Bigl[\frac{\psi'(x)}{\psi''(x)}+\frac{\psi'(y)}{\psi''(y)}-\frac{\psi'(x+y)}{\psi''(x+y)}\Bigr]} {[\psi'(x)\psi'(x+y)+\psi'(y)\psi'(x+y)-\psi'(x)\psi'(y)]^2}, \quad x,y\in M.

— The decrease in sectional curvature of the Fisher--Rao manifold of beta distributions  (2609.10628 - Qi, 9 Sep 2026) in Introduction, Conjecture 1 (labeled Conjecture \ref{Alice-lower-bound-conj}); see also the discussion immediately preceding the conjecture