Log-Concavity of Conic Intrinsic Volumes

What the math gives to ML

The paper supplies a dimension-aware shape constraint for the Gaussian geometry of convex cones: their intrinsic-volume distribution is not merely log-concave, but satisfies an explicit strengthened inequality with dimension-dependent factors. This can be transferred to neural networks whose decisions partition feature space into homogeneous polyhedral cones, especially hard MoE routers and conic classifiers. The most practical use is a regularizer or diagnostic computed from Monte Carlo Gaussian projections, discouraging routers from creating geometrically pathological decision regions while preserving controlled distributions over boundary dimensions.