Quasianalyticity, uncertainty, and integral transforms on higher grassmannians

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We investigate the support of a distribution f on the real grassmannian Grk(Rn) whose spectrum, namely its nontrivial O(n)-components, is restricted to a subset Λ of all O(n)-types. We prove that unless Λ is co-sparse, f cannot be supported at a point. We utilize this uncertainty principle to prove that if 2≤k≤n−2, then the cosine transform of a distribution on the grassmannian cannot be supported inside any single open Schubert cell Σk. The same holds for certain more general α-cosine transforms and for the Radon transform between grassmannians, and more generally for various GLn(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.

Original languageEnglish
Article number109348
JournalAdvances in Mathematics
StatePublished - 15 Dec 2023


FundersFunder number
Israel Science Foundation1750/20


    • Cosine transform
    • Geometric tomography
    • Quasianalytic
    • Radon transform
    • Uncertainty principle
    • Valuations


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