From the Sphere to the Grassmannian: Continuous Brascamp--Lieb Inequalities via Measure Approximation and Heat Flow

We study continuous Brascamp--Lieb inequalities associated with isotropic measures on the sphere $S^{n-1}$ and on the Grassmannian $G(n,k)$. We give two proofs of the continuous Grassmannian inequality. The first proof approximates the continuous isotropic measure by discrete isotropic measures and