Closed-Form Gaussian Mixture Representations of d-Dimensional Triangulated Domains

We derive a Gaussian Mixture representation of a full-dimensional subset of Euclidean space with finite volume from a finite simplicial triangulation in closed form. On each simplex, we replace the uniform probability density function with a Gaussian density that minimizes the Kullback--Leibler divergence, arriving at the information projection onto the family of Gaussian distributions. By exploiting affine isomorphisms between simplices, the joint convexity and bijection invariance properties of the Kullback--Leibler divergence, identities for Dirichlet distributions, and a partition-of-unity argument, we establish a dimension-dependent but geometry-independent sharp upper bound on the global Kullback--Leibler divergence between the resulting Gaussian Mixture and the uniform distribution on the original domain. Numerical examples illustrate the construction and demonstrate its potential in practical applications.