Asymptotic formula for the tail of the maximum of smooth stationary Gaussian fields on non locally convex sets

In this paper we consider the distribution of the maximum of a Gaussian field defined on non locally convex sets. Adler and Taylor or Azaïs and Wschebor give the expansions in the locally convex case. The present paper generalizes their results to the non locally convex case by giving a full expansion in dimension 2 and some generalizations in higher dimension. For a given class of sets, a Steiner formula is established and the correspondence between this formula and the tail of the maximum is proved. The main tool is a recent result of Azaïs and Wschebor that shows that under some conditions the excursion set is close to a ball with a random radius. Examples are given in dimension 2 and higher.