LAGRANGIANS SATISFYING CROFTON FORMULAS, RADON TRANSFORMS, AND NONLOCAL DIFFERENTIALS

This paper introduces a class of geometric objects called Crofton k-de nsities which are the analogue of closed differential forms. In R(n), Crofton densities can be represented by means of a generalization of t he Radon transform. This transform maps functions on the space of (n-k )-planes in R(n) into Crofton k-densities. We write a system of PDE's satisfied by Crofton k-densities. Crofton k-denisites can be considere d as Lagrangians, and we can write multidimensional Euler-Lagrange equ ations for them. The system of PDE's for Crofton densities coincides w ith the lower order terms of the Euler operator. So Euler-Lagrange equ ations for Lagrangians corresponding to Crofton densities have more si mple form than in general. In addition Crofton k-densities can be char acterized by the fact that their extremals include all k-dimensional p lanes. In particular Crofton 1-densities can be considered as Lagrangi ans, and their extremals are straight lines. We present a Crofton 1-de nsity as a ''nonlocal differential'' of a function in R(n). The Poinca re lemma is valid for Crofton 1-densities that satisfy some growth con ditions. In this work we connect two integral geometries, classical in tegral geometry of Poincare and Chern and integral geometry of Radon t ransform. (C) 1994 Academic Press, Inc.