The Problem of Two Sticks

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