On curves contained in convex subsets of the plane

TL;DR

This paper generalizes the perimeter comparison for convex bodies in the plane by establishing a relationship between curve length, convex set parameters, and the number of line intersections, extending classical geometric inequalities.

Contribution

It introduces a new parameter s that bounds the length of curves in convex bodies based on intersection properties, generalizing perimeter inequalities.

Findings

s equals rp/2 when r is even

s equals (r-1)p/2 + d when r is odd

Provides a precise measure for curve lengths intersecting convex bodies

Abstract

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.