We study 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 density F0 on Rd and a time-step 2 h 0, we seek to minimize I(F ) = hS(F )+W2 (F0 , F ) over all of the probability densities F that have the same mean and variance as F0 , where S(F ) is the entropy of F . We prove existence of minimizers. We also analyze the induced geometry of the set of densities satisfying the constraint on.
