Curves with increasing chords in normed planes
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866918134382854144 |
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| author | Lángi, Zsolt Lengyel, Sára |
| author_facet | Lángi, Zsolt Lengyel, Sára |
| contents | 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 $\frac{2π}{3}$ 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2509_02312 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Curves with increasing chords in normed planes Lángi, Zsolt Lengyel, Sára Metric Geometry 52A21, 52A38, 52A40 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 $\frac{2π}{3}$ 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. |
| title | Curves with increasing chords in normed planes |
| topic | Metric Geometry 52A21, 52A38, 52A40 |
| url | https://arxiv.org/abs/2509.02312 |