Convex valuations from Whitney to Nash

Abstract

We consider the Whitney problem for valuations: Does a smooth j-homogeneous translation-invariant valuation on 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 = Gr<inf>r</inf> (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. Informally, the obstruction for a j-density to extend to a j-homogeneous valuation is localized in a single dimension, namely j + 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, 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, answering a question of Fu. As an intermediate step of independent interest, we construct Crofton formulas for all odd translation-invariant valuations.