  We consider the problem of testing, for a given set of planar regions $\cal
R$ and an integer $k$, whether there exists a convex shape whose boundary
intersects at least $k$ regions of $\cal R$. We provide a polynomial time
algorithm for the case where the regions are disjoint line segments with a
constant number of orientations. On the other hand, we show that the problem is
NP-hard when the regions are intersecting axis-aligned rectangles or 3-oriented
line segments. For several natural intermediate classes of shapes (arbitrary
disjoint segments, intersecting 2-oriented segments) the problem remains open.
