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 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.

By: Don Coppersmith; Gyozo Nagy; Sasha Ravsky

Published in: Studia Scientiarum Mathematicarum Hungarica, volume 42, (no 1), pages 107-12 in 2005

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