Lattice problems and norm embeddings

We present reductions from lattice problems in the ℓ₂ norm to the corresponding problems in other norms such as ℓ₁, ℓ_∞ (and in fact in any other ℓ_p norm where 1 ≤ p ≤ ∞). We consider lattice problems such as the Shortest Vector Problem, Shortest Independent Vector Problem, Closest Vector Problem and the Closest Vector Problem with Preprocessing. Most reductions are simple and follow from known constructions of embeddings of normed spaces. Among other things, our reductions imply that the Shortest Vector Problem in the ℓ₁ norm and the Closest Vector Problem with Preprocessing in the ℓ_∞ norm are hard to approximate to within any constant (and beyond). Previously, the former problem was known to be hard to approximate to within 2 - ε, while no hardness result was known for the latter problem.