The Spherical Grünbaum Inequality

What the math gives to ML

The paper gives a sharp, dimension-dependent lower bound on how much spherical convex mass must lie on either side of every hyperplane through the spherical centroid. The transferable asset is a geometric anti-collapse certificate: a nondegenerate set on a hypersphere cannot place less than approximately e^{-1} of its mass in any centroid-tangent hemisphere. This can be converted into a regularizer for hyperspherical embeddings, prototypes, or latent codes that penalizes directional imbalance without choosing fixed coordinate axes. The theorem applies exactly to uniform measure on a spherical convex body, while neural implementations should treat its empirical soft version as a heuristic or diagnostic for arbitrary point clouds.