A Characterization of sets of finite perimeter via their covariogram
Résumé
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. They therefore constitute a characterization of sets of finite perimeter and of sets of finite directional variation. As a consequence we prove 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 formulas 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.
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