A differential equation related to the $\mathbf{l}^{p}$-norms
Volume 101 / 2011
Annales Polonici Mathematici 101 (2011), 251-265
MSC: Primary 46B20, 46B45; Secondary 34A34.
DOI: 10.4064/ap101-3-5
Abstract
Let $p\in (1,\infty )$. The question of existence of a curve in $\mathbb{R}% _{+}^{2}$ starting at $(0,0)$ and such that at every point $(x,y)$ of this curve, the $\mathbf{l}^{p}$-distance of the points $(x,y)$ and $(0,0)$ is equal to the Euclidean length of the arc of this curve between these points is considered. This problem reduces to a nonlinear differential equation. The existence and uniqueness of solutions is proved and nonelementary explicit solutions are given.
