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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?

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There is a paper here:

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.

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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 '11 at 23:42

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.

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