  The lattice size of a lattice polytope $P$ was defined and studied by
Schicho, and Castryck and Cools. They provided an "onion skins" algorithm for
computing the lattice size of a lattice polygon $P$ in $\mathbb{R}^2$ based on
passing successively to the convex hull of the interior lattice points of $P$.
  We explain the connection of the lattice size to the successive minima of
$K=\left(P+(-P)\right)^\ast$ and to the lattice reduction with respect to the
general norm that corresponds to $K$. It follows that the generalized Gauss
algorithm of Kaib and Schnorr (which is faster than the "onion skins"
algorithm) computes the lattice size of any convex body in $\mathbb{R}^2$.
  We extend the work of Kaib and Schnorr to dimension 3, providing a fast
algorithm for lattice reduction with respect to the general norm defined by a
convex origin-symmetric body $K\subset\mathbb{R}^3$. We also explain how to
recover the successive minima of $K$ and the lattice size of $P$ from the
obtained reduced basis and therefore provide a fast algorithm for computing the
lattice size of any convex body $P\subset\mathbb{R}^3$.
