Bicovariograms and Euler characteristic of regular sets

We establish an expression of the Euler characteristic of a r ‐regular planar set in function of some variographic quantities. The usualC 2 framework is relaxed to aC 1 , 1regularity assumption, generalising existing local formulas for the Euler characteristic. We give also general bounds on the number of connected components of a measurable set of R 2 in terms of local quantities. These results are then combined to yield a new expression of the mean Euler characteristic of a random regular set, depending solely on the third order marginals for arbitrarily close arguments. We derive results for level sets of some moving average processes and for the boolean model with non‐connected polyrectangular grains in R 2 . Applications to excursions of smooth bivariate random fields are derived in the companion paper [25][R. Lachièze‐Rey, ], and applied for instance toC 1 , 1Gaussian fields, generalising standard results.