The Problem of Two Sticks

Abstract

Let $l=[l_0,l_1]$ be the directed line segment from $l_0\in {\mathbb{R}}^n$ to $l_1\in {\mathbb{R}}^n.$ Suppose $\bar{l}=[{\bar{l}}_0,{\bar{l}}_1]$ is a second segment of equal length such that $l,\bar{l}$ satisfy the "two sticks condition": $\Vert l_1-{\bar{l}}_0\Vert \ge \Vert l_1-l_0\Vert ,\Vert {\bar{l}}_1-l_0\Vert \ge \Vert {\bar{l}}_1-{\bar{l}}_0\Vert .$ Here $\Vert \cdot \Vert$ is a norm on ${\mathbb{R}}^n.$ We explore the manner in which $l_1-{\bar{l}}_1$ is then constrained when assumptions are made about "intermediate points" $l_{\ast }\in l$, ${\bar{l}}_{\ast }\in \bar{l}.$ Roughly speaking, our most subtle result constructs parallel planes separated by a distance comparable to $\Vert l_{\ast }-{\bar{l}}_{\ast }\Vert$ such that $l_1-{\bar{l}}_1$ must lie between these planes, provided that $\Vert \cdot \Vert$ is "geometrically convex" and "balanced", as defined herein. The standard $p$-norms are shown to be geometrically convex and balanced. Other results estimate $\Vert l_1-{\bar{l}}_1\Vert$ in a Lipschitz or Hölder manner by $\Vert l_{\ast }-{\bar{l}}_{\ast }\Vert$. All these results have implications in the theory of eikonal equations, from which this "problem of two sticks" arose.

AMSLaTeX, 34 pages