The problem of two sticks

Let l=[l0,l1]l=[l0,l1] be the directed line segment from l0∈Rnl0∈Rn to l1∈Rnl1∈Rn. Suppose that 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 RnRn. We explore the manner in which l1,l̄1 are then constrained when assumptions are made about “intermediate points” lt∈l,l̄t∈l̄. Roughly speaking, our most subtle result constructs parallel planes separated by a distance comparable to ‖lt−l̄t‖ and whose normal is nearly orthogonal to l̄1−l̄0 such that both l1l1 and l̄1 must lie between these planes, provided that ‖⋅‖‖⋅‖ is “geometrically convex” and “balanced”, as defined herein. Moreover, given a family of “sticks” which pairwise satisfy the two sticks condition, all with intermediate points in a fixed small ball, the planes can be chosen to contain the terminal points of all the sticks in the family. The standard pp-norms are shown to be geometrically convex and balanced. Other results estimate ‖l1−l̄1‖ in a Lipschitz or Hölder manner by ‖lt−l̄t‖. All these results have implications in the theory of eikonal equations, from which this “problem of two sticks” arose.