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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.

Willie Wong
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begeistzwerst
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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
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    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
  • Gardner does not really treat the "Crofton intersection formulas", as he calls them, but only mentions them in the Notes to Chapter 7. However, he lists some related work in his huge bibliography, so there's a chance that I'll find what I am looking for. Thanks for the hint! – begeistzwerst May 23 '12 at 09:09
  • Maybe you can have a look a the really complete book: "Stochastic and integral geometry" from Schneider and Weil. – Gilles Bonnet May 09 '14 at 15:35

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