Quasianalyticity, uncertainty, and integral transforms on higher grassmannians

TL;DR

This paper establishes an uncertainty principle for distributions on real grassmannians, showing restrictions on their support based on spectral components, and applies these results to improve classical theorems in convex geometry and geometric tomography.

Contribution

It introduces a new uncertainty principle for distributions on grassmannians and applies it to derive sharper results in convex geometry and geometric tomography.

Findings

Distributions with restricted spectrum cannot be supported at a point unless co-sparse.

Cosine transforms on grassmannians cannot be supported in a single Schubert cell.

Sharper versions of classical theorems in convex geometry and tomography are obtained.

Abstract

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.