  We present a new combinatorial formula for Hall-Littlewood functions
associated with the affine root system of type $\tilde A_{n-1}$, i.e.
corresponding to the affine Lie algebra $\hat{\mathfrak{sl}}_n$. Our formula
has 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 points
in a certain infinite-dimensional convex polyhedron. We derive a weighted
version of Brion's theorem and then apply it to our polyhedron to prove the
formula.
