  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": $\| l_1-\bar l_0\| \ge \| l_1-l_0\|, \| \bar l_1-l_0\| \ge \| \bar
l_1-\bar l_0\|.$ 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 H\"older manner by $\| l_* -\bar l_* \| $. All these results
have implications in the theory of eikonal equations, from which this "problem
of two sticks" arose.
