  Let $B$ be a Borel set in $\mathbb E^{d}$ with volume $V(B)=\infty$. It is
shown that almost all lattices $L$ in $\mathbb E^{d}$ contain infinitely many
pairwise disjoint $d$-tuples, that is sets of $d$ linearly independent points
in $B$. A consequence of this result is the following: let $S$ be a star body
in $\mathbb E^{d}$ with $V(S)=\infty$. Then for almost all lattices $L$ in
$\mathbb E^{d}$ the successive minima $\lambda_{1}(S,L),..., \lambda_{d}(S,L)$
of $S$ with respect to $L$ are 0. A corresponding result holds for most
lattices in the Baire category sense. A tool for the latter result is the
semi-continuity of the successive minima.
