  Minkowski's First Theorem and Dirichlet's Approximation Theorem provide upper
bounds on certain minima taken over lattice points contained in domains of
Euclidean spaces. We study the distribution of such minima and show, under some
technical conditions, that they exhibit Weibull asymptotics with respect to
different natural measures on the space of unimodular lattices in $\bR^d$. This
follows from very general Poisson approximation results for shrinking targets
which should be of independent interest. Furthermore, we show in the appendix
that the logarithm laws of Kleinbock-Margulis, Khinchin and Gallagher can be
deduced from our distributional results.
