On curves contained in convex subsets of the plane
  If K' and K are convex bodies of the plane such that K' is a subset of K then the perimeter of K' is not greater than the perimeter of K. We obtain the following generalization of this fact. Let K be a convex compact body of the plane with the perimeter p and the diameter d and r>1 be an integer. Let s be the smallest number such that for any curve of length greater than s contained in K there is a straight line intersecting the curve at least in r+1 different points. Then s=rp/2 if r is even and s=(r-1)p/2+d if r is odd. 
Don Coppersmith, Gyozo Nagy and Alex Ravsky
math.MG