Let L be a lattice in R^n. Suppose that L has the property that the determinant of every non-zero sublattice L' of L is larger than one. We prove that there exists a symmetric random vector X on R^n with a smooth, strictly positive density f on R^n simultaneously enjoying the following two properties: a) f is strongly log-concave; b) X + L is uniform on R^n/L.

Moreover, the existence of such a density f on R^n characterizes this determinant condition imposed on L. This construction connects several problems in lattice theory to established results in convex geometry. We discuss a range of applications. This is joint work with Victor Reis and Thomas Rothvoss.