Introduction

In this chapter we investigate the geometry of the surface at distance ε from the convex hull boundary. We show that the nearest point map induces a bilipschitz homeomorphism between an open simply connected domain on the 2-sphere at infinity and the associated convex hull boundary. After that we show how to extend a lamination on the hyperbolic plane to a foliation in an open neighbourhood of the support of the lamination. By averaging the nearest point map along leaves of the orthogonal foliation, one obtains a bilipschitz homeomorphism ρ between the ε distant surface and the convex hull boundary. The properties of ρ are determined by means of a careful investigation of the foliation, and the properties of the foliation are determined by progressively complicating the situation. The crucial lemmas are given similar names in the progressively more complicated situations, in order to make the parallels clearer.

The epsilon distant surface

The proof of Sullivan's theorem, relating the convex hull boundary with the surface at infinity, is carried out by factoring through Sε, the surface at distance ε from the convex hull boundary component S. For this reason, we need to know quite a lot about Sε. In this section we investigate its geometry, in the case of a finitely bent convex hull boundary component. We analyze the structure of geodesics and show that the distance function is C1.

Let ∧ be a closed subset of S2, whose complement is a topological disk. Let S be the boundary of C(∧).