Excursions above high levels by Gaussian random fields

For a two-dimensional, homogeneous, Gaussian random field X(t) and compact, convex S [subset of] R2 we show that as u --> [infinity] the set Au = {t [set membership, variant] S : X(t) [greater-or-equal, slanted] u} possesses, with probabilityapproaching one, components that are approximately convex. Furthermore, the function X is also approximately concave over Au. One of the main aims of the paper is, at the cost of losing some detail, to simplify the analytic complexity of previous results about high level excursions of Gaussian fields by judicious use of concepts from integral and differential geometry.