Title: On curves contained in convex subsets of the plane
Abstract: If K' ? K are convex bodies of the plane 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 perimeter p and diameter d and let 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.