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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 $p=1$.

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This paper by Elkies, Odlyzko, and Rush gives a bound in the other direction, namely a lower bound for the packing density of $L_p$ balls: dx.doi.org/10.1007/BF01232282 . Hopefully Noam will stop by and tell us whether there is an easy upper bound. –  Yoav Kallus May 29 '13 at 0:11
    
This is interesting, thanks for the reference Yoav! –  Marcel Celaya May 29 '13 at 19:37
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