2025/09/02 by Zsolt Lángi, Lángi, Zsolt, Sára Lengyel +1
Computer Science · Mathematics · #52A21 #52A38 #52A40 #Computational Geometry and Mesh Generation #FOS: Mathematics #Limits and Structures in Graph Theory #Metric Geometry (math.MG) #Point processes and geometric inequalities
paper · pdf · doi:10.48550/arxiv.2509.02312
openalex publication_date 2025/09/02 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 (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.