A Geometric-Mean Proof of the Spherical Simplex Volume-Product Bound

The product of the normalized solid angles of a simplicial cone in \(\mathbb R^n\) and its positive dual is at most \(4^{-n}\), with equality exactly for orthogonal images of the positive orthant. This note gives a short proof by applying the classical geometric-mean Prékopa–Leindler inequality to two inverse positive definite quadratic forms. The result answers the upper-bound question in Problem 23 of the 2010 AIM workshop list on Mahler’s conjecture and duality, corresponding to UnsolvedMath v1.6.0 record AIM-CONVEX_GEOMETRY-0019. The functional inequality and its equality criterion are credited to established literature; the spherical conclusion is a direct application, and no historical priority claim is made. An explicit second variation in unit-diagonal Gram coordinates gives a positive quadratic deficit and a dimension-dependent local stability estimate. The accompanying code supplies symbolic identities and numerical sanity checks, not a formal proof. This is an unrefereed English preprint.