  We quantify the large deviations of Gaussian extreme value statistics on
closed convex sets in d-dimensional Euclidean space. The asymptotics imply that
the extreme value distribution exhibits a rate function that is a simple
quadratic function of a unique "dominating point" located on the boundary of
the convex set. Furthermore, the dominating point is identified as the
optimizer of a certain convex quadratic programming problem, indicating a
"collusion" between the dependence structure of the Gaussian random vectors and
the geometry of the convex set in determining the asymptotics. We specialize
our main result to polyhedral sets which appear frequently in other contexts
involving logarithmic asymptotics. We also extend the main result to
characterize the large deviations of Gaussian-mixture extreme value statistics
on general convex sets. Our results have implications to contexts arising in
rare-event probability estimation and stochastic optimization, since the nature
of the dominating point and the rate function suggest importance sampling
measures.
