Voronoi Cells of Lattices with Respect to Arbitrary Norms

We study the geometry and complexity of Voronoi cells of lattices with respect to arbitrary norms. On the positive side, we show for strictly convex and smooth norms that the geometry of Voronoi cells of lattices in any dimension is similar to the Euclidean case, i.e., the Voronoi cells are defined by the Voronoi-relevant vectors and the facets of a Voronoi cell are in one-to-one correspondence with these vectors. On the negative side, we show that Voronoi cells are combinatorially much more complicated for arbitrary strictly convex and smooth norms than in the Euclidean case. In particular, we construct a family of three-dimensional lattices whose number of Voronoi-relevant vectors with respect to the $\ell_3$-norm is unbounded. Our result indicates that the single exponential time algorithm of Micciancio and Voulgaris for solving the shortest vector problem (SVP) and the closest vector problem (CVP) in the Euclidean norm cannot be extended to achieve deterministic single exponential time algorithms for lattice problems with respect to arbitrary $\ell_p$-norms. In fact, the algorithm of Micciancio and Voulgaris and its run time analysis crucially depend on the fact that, for the Euclidean norm, the number of Voronoi-relevant vectors is single exponential in the lattice dimension.