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I am looking for a good introductory reference (book, lecture notes, survey article) on integral geometry. I am especially interested in the Crofton formula in $\mathbb{R}^n$ and its extensions to Riemannian manifolds.

So far, I have only had a look at the 2004 edition of Santaló's integral geometry and geometric probability and Blaschke's "Vorlesungen über Integralgeometrie" from 1955. I find both of them rather inaccessible (for someone not acquainted with the field like me). I also found this PDF book by Rémi Langevin, which happens to be quite meagre on the topics I am interested in.

I am thankful for any suggestions.

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    Maybe you can have a look a the really complete book: "Stochastic and integral geometry" from Schneider and Weil.2014-05-09

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