Minkowski's theorem states that if $K\subseteq\mathbb{Z}^n$ is a convex compact set, $K=-K$, and $\mathrm{volume}(K)\geq 2^n$, then $K$ contains a nonzero integral vector.

Can this bound be improved in the special case when $K$ is the image of the unit $L_p$ ball under a (full-dimensional) linear transformation, for some fixed $p$? I am particularly interested the case $p=1$.