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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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Another classics on the subject: Herbert Solomon, Geometric Probability, SIAM, 1978. – Did May 22 '12 at 13:05
Do you know to what extent Solomon treats the Crofton formulae? – begeistzwerst May 22 '12 at 13:21
Chapter 5 is titled Crofton's Theorem and Sylvester's Problem in Two and Three Dimensions hence my guess is he does. – Did May 22 '12 at 13:34
I just had a look at this chapter. I suspect it is about a different theorem by Crofton. Or I completely fail to see the connection... – begeistzwerst May 22 '12 at 15:05
Sorry if the reference is misleading, I do not have the book at hand right now. Another reference I heard experts such as Molchanov or Calka mention is Geometric tomography by R.J. Gardner (but this one I never did even open hence this is mere hearsay...). – Did May 22 '12 at 16:26

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