Distances on Convex Bodies, Cones, and Simplicial Complexes

Abstract

A convex body in the n-dimensional Euclidean space $\mathbb{E}^{n}$ is a convex compact connected subset of $\mathbb{E}^{n}$. It is called solid (or proper) if it has nonempty interior. Let K denote the space of all convex bodies in $\mathbb{E}^{n}$, and let K _p be the subspace of all proper convex bodies. Given a set $X \subset \mathbb{E}^{n}$, its convex hull c o n v(X) is the minimal convex set containing X.