On curves contained in convex subsets of the plane

Author: Coppersmith Don   Nagy Győző   Ravsky Sasha  

Publisher: Akademiai Kiado

ISSN: 0081-6906

Source: Studia Scientiarum Mathematicarum Hungarica, Vol.42, Iss.1, 2005-02, pp. : 107-112

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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.