Leírás
We say that a curve $\gamma$ satisfies the increasing chords property, if for any points $a,b,c,d$ in this order on $\gamma$, the distance of $a,d$ is not smaller than the distance of $b,c$. Binmore asked the question in 1971 if there is a universal constant $C$ such that for any curve $\gamma$ in the Euclidean plane, satisfying the increasing chords property, if the endpoints of $\gamma$ are at unit distance apart, then the arclength of $\gamma$ is at most $C$.
Larman and McMullen showed in 1972 that the constant $C=2\sqrt{3}$ satisfies this condition. Rote proved in 1991 that the optimal such constant is equal to $\frac{2\pi}{3}$.
In this note we give an estimate for the arclengths of curves with the increasing chords property in Euclidean $d$-space, and generalize Rote's result for such curves in a normed plane with a strictly convex norm. Joint work with Adrian Dumitrescu and Sara Lengyel.