The Brunn--Minkowski inequality for the Gaussian measure

What the math gives to ML

The paper gives a sharp, dimension-dependent power-concavity law for Gaussian measure under Minkowski interpolation of convex bodies containing the origin. The transferable asset is not merely the inequality itself, but the explicit exponent and the fact that Gaussian measure of an interpolated convex set is guaranteed by endpoint measures, providing a principled replacement for heuristic latent-space mixing assumptions. A practical neural-network use is to construct convex latent supports for classes or concepts and train their Gaussian mass to obey this inequality under interpolation. This yields a geometry-aware regularizer that can be tested in classifiers or generative encoders, although its guarantees apply only when the learned sets are approximately convex and origin-containing.