  We study the lengths of curves passing through a fixed number of points on
the boundary of a convex shape in the plane. We show that for any convex shape
$K$, there exist four points on the boundary of $K$ such that the length of any
curve passing through these points is at least half of the perimeter of $K$. It
is also shown that the same statement does not remain valid with the additional
constraint that the points are extreme points of $K$. Moreover, the factor
$\frac12$ cannot be achieved with any fixed number of extreme points. We
conclude the paper with few other inequalities related to the perimeter of a
convex shape.
