We study the following problem: Given a continuous domain Omega along with its convex hull K, a point A is an element of K, and a measure mu on Omega, find the probability density over Omega whose marginal is A and that minimizes the KL divergence to the uniform density with respect to mu. Several distributions in mathematics, physics, statistics, and theoretical computer science arise by different settings of the parameters of this problem. We give a polynomial bound on the norm of the optimizer of the dual problem that holds in a very general setting and relies on a "balance" property of the measure mu on Omega, and exact algorithms for evaluating the dual and its gradient for several interesting settings of Omega and mu. Together, along with the ellipsoid method, these results imply polynomial-time algorithms to compute such KL divergence minimizing distributions in several cases. Applications of our results include (1) an optimization characterization of the Goemans-Williamson measure [M. X. Goemans and D. P. Williamson, J ACM, 42 (1995), pp. 1115-1145] that is used to round a positive semidefinite matrix to a vector; (2) the computability of the entropic barrier for convex bodies, given a strong integration oracle, studied by [S. Bubeck and R. Eldan, Proc. Mach. Learn. Res. (PMLR), 40 (2015), p. 279], and (3) a polynomial-time algorithm to compute the barycentric quantum entropy of a density matrix that was proposed as an alternative to von Neumann entropy [W. Band and J. L. Park, Found. Phys., 6 (1976), pp. 249-262; J. L. Park and W. Band, Found. Phys., 7 (1977), pp. 233-244; P. B. Slater, Phys. Lett. A, 159 (1991), pp. 411-414]; this corresponds to the case when Omega is the set of rank-one projection matrices and mu is derived from the Haar measure on the unit sphere.