ON curves contained in convex subsets of the plane

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. © 2005 Akadémiai Kiadó, Budapest.