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Cauchy-Crofton formula for the density of subanalytic sets. (Formule de Cauchy-Crofton pour la densité des ensembles sous-analytiques.) (French) Zbl 0945.32019
Let \(X\) be a bounded \(R\)-dimensional sub-analytic subset of \(\mathbb{R}^n\) and \(\Gamma\) the germ of \(X\) at 0. Let \(\Theta_k (\Gamma,0)\) be the \(k\)-density (or Lelong number) of \(\Gamma\) at 0: \(\Theta_k (\Gamma,0)\) is the limit of \({\mathfrak H}^k (\Gamma\cap B^n_{(0,r)})/{\mathfrak H}^k(B^k_{0,r)})\), \({\mathfrak H}^k\) being the \(k\)-dimensional Hausdorff measure and \(B^\ell_{(0,r)}\) the ball in \(\mathbb{R}^\ell\) of radius \(r\) centered at 0.
The author gives for \(\Theta_k (\Gamma,0)\) an integral formula which is the analogous of the classical Cauchy-Grofton formula for the volume.

32U25 Lelong numbers
14P15 Real-analytic and semi-analytic sets
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