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In this paper we consider the distribution of the maximum of a Gaussian fielddefined on non locally convex sets. Adler and Taylor or Aza\"\i s and Wscheborgive the expansions in the locally convex case. The present paper generalizestheir results to the non locally convex case by giving a full expansion indimension 2 and some generalizations in higher dimension. For a given class ofsets, a Steiner formula is established and the correspondence between thisformula and the tail of the maximum is proved. The main tool is a recent resultof Aza\"\i s and Wschebor that shows that under some conditions the excursionset is close to a ball with a random radius. Examples are given in dimension 2and higher.

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