Computation of the Perimeter of Measurable Sets via their Covariogram. Applications to Random Sets

Bruno Galerne (1)

1 CMLA - Centre de Mathématiques et de Leurs Applications

The covariogram of a measurable set $A$ is the function $g_A$ which to each $y\in\R^d$ associates the Lebesgue measure of $A\cap y+A$. This paper proves two formulas. The first equates the directional derivatives at the origin of $g_A$ to the directional variations of $A$. The second equates the average directional derivative at the origin of $g_A$ to the perimeter of $A$. These formulas, previously known with restrictions, are proved for any measurable set. As a by-product, it is proved that the covariogram of a set $A$ is Lipschitz if and only if $A$ has finite perimeter, the Lipschitz constant being half the maximal directional variation. The two formulas have counterparts for mean covariogram of random sets. They also permit to compute the expected perimeter per unit volume of any stationary random closed set. As an illustration, the expected perimeter per unit volume of homogeneous Boolean models having any grain distribution is computed.

Mots clés: Boolean model; Covariogram; Directional variation; Random closed sets; Set covariance; Set of finite perimeter; Specific variation

Domaines: Analyse classique [math.CA]; Probabilités [math.PR]; Analyse fonctionnelle [math.FA]

Bruno Galerne. Computation of the Perimeter of Measurable Sets via their Covariogram. Applications to Random Sets. 2010. ⟨hal-00480825v2⟩