  Many polytopes arising in polyhedral combinatorics are linear projections of
higher-dimensional polytopes with significantly fewer facets. Such lifts may
yield compressed representations of polytopes, which are typically used to
construct small-size linear programs. Motivated by algorithmic implications for
the closest vector problem, we study lifts of Voronoi cells of lattices.
  We construct an explicit $d$-dimensional lattice such that every lift of the
respective Voronoi cell has $2^{\Omega(d / \log d)}$ facets. On the positive
side, we show that Voronoi cells of $d$-dimensional root lattices and their
dual lattices have lifts with $O(d)$ and $O(d \log d)$ facets, respectively. We
obtain similar results for spectrahedral lifts.
