Abstract Westudy several constrained variational problems in the 2-Wasserstein metric for which the set of probability densities satisfying the constraint is not closed. For example, given a probability densityF0onRd and a time-step h>0, we seek to minimizeI(F)=hS(F)+W22 (F0,F)overall of the probability densities Fthat have the same mean and variance asF0, whereS(F)isthe entropy ofF.Weprove existence of minimizers. We also analyze the induced geometry of the set of densities satisfying the constraint on the variance and means, and we determine all of the geodesics on it. From this, we determine acriterion for convexity of functionals in the induced geometry. It turns out, for example, that the entropy is uniformly strictly convex on the constrained manifold, though not uniformly convex without the constraint. The problems solved here arose in a study of a variational approach to constructing and studying solutions of the nonlinear kinetic Fokker-Planck equation, which is briefly described here and fully developed in a companion paper. Contents 1. Introduction 2. Riemannian geometry of the 2-Wasserstein metric 3. Geometry of the constraint manifold 4. The Euler-Lagrange equation 5. Existence of minimizers References