# zbMATH — the first resource for mathematics

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.

##### MSC:
 32U25 Lelong numbers 14P15 Real-analytic and semi-analytic sets
Full Text: