Crofton Densities and Nonlocal Differentials
This paper introduces a class of geometric objects called Crofton k- densities, which are the analogue of closed differential forms. We define a “nonlocal differential” of a function in R n and prove that it is a Crofton 1-density. The Poincaré lemma is valid for Crofton 1-densities that satisfy some growth conditions.
In R n, Crofton densities can be represented by means of a generalization of the Radon transform. This transform maps functions on the space of (n — k)-planes in R n into Crofton k-densities. Crofton 1-densities can be considered as Lagrangians, and their extremals are straight lines. In the continuation of this paper we shall discuss densities which are Crofton with respect to multiparametric families of curves or surfaces.
KeywordsManifold Radon Stein
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