2023/06/12 by Dmitry Faifman, Faifman, Dmitry, Georg C. Hofstätter +1 · 1 citation
Mathematics · #14N20 #44A15 #52B45 #53A07 #53C65 #Differential Geometry (math.DG) #FOS: Mathematics #Functional Analysis (math.FA) #Mathematics and Applications #Metric Geometry (math.MG) #Point processes and geometric inequalities
paper · pdf · doi:10.48550/arxiv.2306.07390
openalex publication_date 2023/06/12 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We consider the Whitney problem for valuations: does a smooth j-homogeneous translation-invariant valuation on \mathbb Rn exist that has given restrictions to a fixed family S of linear subspaces? A necessary condition is compatibility: the given valuations must coincide on intersections. We show that for S=Grr(\mathbb Rn), the grassmannian of r-planes, this condition becomes sufficient once r≥ j+2. This complements the Klain and Schneider uniqueness theorems with an existence statement, and provides a recursive description of the image of the cosine transform. Informally speaking, we show that the transition from densities to valuations is localized to codimension 2. We then look for conditions on S when compatibility is also sufficient for extensibility, in two distinct regimes: finite arrangements of subspaces, and compact submanifolds of the grassmannian. In both regimes we find unexpected flexibility. As a consequence of the submanifold regime, we prove a Nash-type theorem for valuations on compact manifolds, from which in turn we deduce the existence of Crofton formulas for all smooth valuations on manifolds. As an intermediate step of independent interest, we construct Crofton formulas for all odd translation-invariant valuations.