TITLE: The relationship between Crofton formula and Radon transform.
QUESTION [9 upvotes]: 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?

REPLY [3 votes]: 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.