The famous Crofton formula says that the length of a curve can be calculated by integral of the `line crossing' over the space of all oriented lines. My question is, is there a way to treat this formula as a special case or corollary of the Radon transform theory? If so, how can we express the relationship precisely?
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There is a paper here: http://www.math.poly.edu/~alvarez/pdfs/crofton.pdf that develops a theory of "Gelfand Transforms" which in a sense made precise there is a generalization of both the Radon Transform and the CauchyCrofton formula. 


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