  We consider a natural variation of the concept of stabbing a segment by a
simple polygon: a segment is stabbed by a simple polygon $\mathcal{P}$ if at
least one of its two endpoints is contained in $\mathcal{P}$. A segment set $S$
is stabbed by $\mathcal{P}$ if every segment of $S$ is stabbed by
$\mathcal{P}$. We show that if $S$ is a set of pairwise disjoint segments, the
problem of computing the minimum perimeter polygon stabbing $S$ can be solved
in polynomial time. We also prove that for general segments the problem is
NP-hard. Further, an adaptation of our polynomial-time algorithm solves an open
problem posed by L\"offler and van Kreveld [Algorithmica 56(2), 236--269
(2010)] about finding a maximum perimeter convex hull for a set of imprecise
points modeled as line segments.
