Crofton Densities and Nonlocal Differentials

  • I. M. Gelfand
  • M. M. Smirnov


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 (nk)-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.


Dual Function Inversion Formula Length Element Integral Geometry Dual Measure 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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Copyright information

© Springer Science+Business Media New York 1993

Authors and Affiliations

  • I. M. Gelfand
    • 1
    • 2
  • M. M. Smirnov
    • 1
    • 2
  1. 1.Mathematics DepartmentRutgers UniversityNew BrunswickUSA
  2. 2.Mathematics DepartmentPrinceton UniversityPrincetonUSA

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