标题： 具有递增弦的赋范平面中的曲线 显示英文标题

标题： Curves with increasing chords in normed planes

摘要： 如果对于曲线上按顺序排列的任意点$a,b,c,d$，点$a,d$的距离不小于点$b,c$的距离，则该曲线具有递增弦性质。 回答 Larman 和 McMullen 的一个猜想，Rote 在 1994 年证明了具有递增弦性质的欧几里得平面上的曲线的弧长最多是其端点距离的$\frac{2\pi}{3}$倍，且这个不等式是紧的。 在本文中，我们基于对赋范平面中渐开线几何性质的研究，将 Rote 的结果推广到具有严格凸范数的赋范平面上的曲线。 我们还讨论了一些相关的极值问题。 显示英文摘要

摘要： 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.