Preprint Version: May 30, 2009 THE PROBLEM OF TWO STICKS

Let l = [l0, l1] be the directed line segment from l0 ∈ IR to l1 ∈ IR. Suppose l̄ = [l̄0, l̄1] is a second segment of equal length such that l, l̄ satisfy the “two sticks condition”: ∥∥l1 − l̄0∥∥ ≥ ‖l1 − l0‖ ,∥∥l̄1 − l0∥∥ ≥ ∥∥l̄1 − l̄0∥∥ . Here ‖·‖ is a norm on IR. We explore the manner in which l1− l̄1 is then constrained when assumptions are made about “intermediate points” l∗ ∈ l, l̄∗ ∈ l̄. Roughly speaking, our most subtle result constructs parallel planes separated by a distance comparable to ∥∥l∗ − l̄∗∥∥ such that l1 − l̄1 must lie between these planes, provided that ‖·‖ is “geometrically convex” and “balanced”, as defined herein. The standard p-norms are shown to be geometrically convex and balanced. Other results estimate ∥∥l1 − l̄1∥∥ in a Lipschitz or Hölder manner by ∥∥l∗ − l̄∗∥∥. All these results have implications in the theory of eikonal equations, from which this “problem of two sticks” arose. Introduction The origin of the “problem of two sticks,” which we are about to describe, lies in the theory of eikonal equations. Roughly speaking, the results of Caffarelli and Crandall [3] rely on knowledge of how the endpoints of “rays” of the distance function to some set, as measured in a norm ‖·‖ , that emanate from points in the set and pass through a common tiny ball in the interior of the region of differentiability of the distance function are constrained. We provide a variety of results that speak to this issue. In particular, the crown jewel of our results, Corollary 5.3 below, implies that the endpoints must lie between parallel planes which are separated by a distance comparable to the radius of the ball. The ingredients of the problem of two sticks are a norm ‖·‖ on IR and two “sticks” l = [l0, l1], l̄ = [l̄0, l̄1], where [l0, l1] denotes the directed line segment from l0 to l1 ∈ IR. Sometimes we regard l as a set, as when we write x ∈ l, or x ∈ [l0, l1], but [x, y] has an “initial” point x and a “terminal” point y. We assume throughout this paper that the sticks satisfy the “two sticks