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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?

flag	asked Mar 24 2011 at 21:54 David 103●3

Liviu Nicolaescu · Answer 1 · 2012-02-10 14:51:40Z UTC

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You should definitely check these notes generated by three bright undergraduates for an REU project that I supervised a few years ago. I promise you, it will be worth your time.

answered Feb 10 2012 at 14:51

Liviu Nicolaescu
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Dick Palais · Answer 2 · 2011-03-24 22:21:05Z UTC

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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 Cauchy-Crofton formula.

answered Mar 24 2011 at 22:21

Dick Palais
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Thanks Dick. That seems to be a beautiful generalization. However it's somehow beyond my expectation, which is trying to find some insight of the Crofton formula by means of Radon transform, instead of lifting both of them to another level. Anyway, thanks a lot for the recommendation of the paper. – David Mar 24 2011 at 23:42

The relationship between Crofton formula and Radon transform.

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