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arXiv · 2509.02312

Curves with increasing chords in normed planes

Zsolt Lángi·Sára Lengyel
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Abstract

A curve has the increasing chord property if for any points $a,b,c,d$ in this order on the curve, the distance of $a,d$ is not smaller than that of $b,c$. Answering a conjecture of Larman and McMullen, Rote proved in 1994 that the arclength of a curve in the Euclidean plane with the increasing chord property is at most $\frac{2\pi}{3}$ times the distance of its endpoints, and this inequality is sharp. In this note we generalize the result of Rote for curves in a normed plane with a strictly convex norm, based on an investigation of the geometric properties of involutes in normed planes. We also discuss some related extremum problems.

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Zsolt Lángi, Sára Lengyel. 2025-09-02. Curves with increasing chords in normed planes. https://arxiv.org/abs/2509.02312

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