Chapter II.1 - Convex hulls

Summary

Introduction

In this chapter we discuss the geometric properties of convex subsets of ℍ³. In particular we discuss the hyperbolic convex hull of a closed subset of S², regarded as the boundary of ℍ³. We show that, with respect to the metric induced by the length of rectifiable paths, the boundary of such a convex hull is a complete hyperbolic 2-manifold. Following (Thurston, 1979) we show that we obtain a measured lamination from the boundary, where the measure tells one how much the surface is bent.

Hyperbolic convex hulls

We will consider the open unit ball in ℝⁿ as the hyperbolic space ℍⁿ, giving it the Poincaré metric 2dr/(1 - r²), where r is the euclidean distance to the origin and dr is the euclidean distance element. Hyperbolic isometries act on the closed unit ball conformally. Thus we get hyperbolic geometry inside the ball and conformal geometry on the boundary S^n-1. We are mainly interested in the case n = 3, though we will also need to discuss n = 2 from time to time. We denote the closed unit ball, with its conformal structure and with the hyperbolic structure on its interior, by Bⁿ.

Definition. A non-empty subset X of Bⁿ is said to be convex or, more precisely, hyperbolically convex if, given any two points of X, the geodesic arc joining them also lies in X.