Quasianalyticity, uncertainty, and integral transforms on higher grassmannians

We investigate the support of a distribution $f$ on the real grassmannian $\mathrm{Gr}_k(\mathbb R^n)$ whose spectrum, namely its nontrivial $\mathrm O(n)$-components, is restricted to a subset $\Lambda$ of all $\mathrm O(n)$-types. We prove that unless $\Lambda$ is co-sparse, $f$ cannot be supported at a point. We utilize this uncertainty principle to prove that if $2\leq k\leq n-2$, then the cosine transform of a distribution on the grassmannian cannot be supported inside any single open Schubert cell $\Sigma^k$. The same holds for certain more general $\alpha$-cosine transforms and for the Radon transform between grassmannians, and more generally for various $\mathrm{GL}_n(\mathbb R)$-modules. These results are then applied to convex geometry and geometric tomography, where sharper versions of the Aleksandrov projection theorem, Funk section theorem, and Klain's and Schneider's injectivity theorems for convex valuations are obtained.