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": $\ l_1\bar l_0\ \ge \ l_1l_0\, \ \bar l_1l_0\ \ge \ \bar l_1\bar l_0\.$ Here $\ \cdot\ $ 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" $l_* \in l$, $\bar l_* \in \bar l.$ Roughly speaking, our most subtle result constructs parallel planes separated by a distance comparable to $\ l_* \bar l_*\ $ such that $l_1\bar l_1$ must lie between these planes, provided that $\ \cdot\ $ is "geometrically convex" and "balanced", as defined herein. The standard $p$norms are shown to be geometrically convex and balanced. Other results estimate $\ l_1\bar l_1 \$ in a Lipschitz or Hölder manner by $\ l_* \bar l_* \ $. All these results have implications in the theory of eikonal equations, from which this "problem of two sticks" arose.
 Publication:

arXiv eprints
 Pub Date:
 January 2010
 DOI:
 10.48550/arXiv.1001.5186
 arXiv:
 arXiv:1001.5186
 Bibcode:
 2010arXiv1001.5186C
 Keywords:

 Mathematics  Differential Geometry;
 51B20;
 52A21
 EPrint:
 AMSLaTeX, 34 pages