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.