Chapter 9 - Distances on Convex Bodies, Cones, and Simplicial Complexes

Abstract

This chapter discusses concepts related to distances on convex bodies, cones, and simplicial complexes. A convex body in the n-dimensional Euclidean space En is a compact convex subset of En. It is called proper if it has nonempty interior. If K denotes the space of all convex bodies in En, and if Kp be the subspace of all proper convex bodies, then any metric space (K, d) on K is called metric space of convex bodies. Metric spaces of convex bodies, in particular the metrization by the Hausdorff metric, or by the symmetric difference metric, play a basic role in the foundations of analysis in Convex Geometry. The Eggleston distance (or symmetric surface area deviation) is a distance on Kp, defined by S(C U D) –S(C ∩ D), where S(·) is the surface area. The measure of surface deviation is not a metric. For convex polyhedra, the growth distance is defined as the amount objects must be grown from their internal seed points until their surfaces touch. The chapter further discusses concepts related to template metric, perimeter deviation, Pompeiu-Eggleston metric, Sobolev distance, and Thompson part metric.