Note on Shortest and Nearest Lattice Vectors

K is a centrally symmetric convex body with nonempty interior and fK(·) is also called the distance function of K because fK(x) = min{ρ ∈ R≥0 : x ∈ ρK}. The Euclidean norm is denoted by fB(·), where B is the n-dimensional unit ball, and the associated inner product is denoted by 〈·, ·〉. Finally, we denote by C the cube with edge length 2 and center 0, and thus fC(·) denotes the maximum norm. As usual we denote by dxe the smallest integer not less than x ∈ R.