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We present a new combinatorial formula for Hall-Littlewood functionsassociated with the affine root system of type $\tilde A_{n-1}$, i.e.corresponding to the affine Lie algebra $\hat{\mathfrak{sl}}_n$. Our formulahas the form of a sum over the elements of a basis constructed by Feigin,Jimbo, Loktev, Miwa and Mukhin in the corresponding irreducible representation.  Our formula can be viewed as a weighted sum of exponentials of integer pointsin a certain infinite-dimensional convex polyhedron. We derive a weightedversion of Brion's theorem and then apply it to our polyhedron to prove theformula.

A Combinatorial Formula for Affine Hall-Littlewood Functions via a Weighted Brion Theorem