  Let $\eta_t$ be a Poisson point process of intensity $t\geq 1$ on some state
space $\Y$ and $f$ be a non-negative symmetric function on $\Y^k$ for some
$k\geq 1$. Applying $f$ to all $k$-tuples of distinct points of $\eta_t$
generates a point process $\xi_t$ on the positive real-half axis. The scaling
limit of $\xi_t$ as $t$ tends to infinity is shown to be a Poisson point
process with explicitly known intensity measure. From this, a limit theorem for
the the $m$-th smallest point of $\xi_t$ is concluded. This is strengthened by
providing a rate of convergence. The technical background includes Wiener-It\^o
chaos decompositions and the Malliavin calculus of variations on the Poisson
space as well as the Chen-Stein method for Poisson approximation. The general
result is accompanied by a number of examples from geometric probability and
stochastic geometry, such as Poisson $k$-flats, Poisson random polytopes,
random geometric graphs and random simplices. They are obtained by combining
the general limit theorem with tools from convex and integral geometry.
