The horofunction boundary of finite-dimensional ${\ell }_p$ spaces

Abstract

We give a complete description of the horofunction boundary of finite-dimensional ${\ell }_p$ spaces for $1\le p\le \infty$. We also study the variation norm on ${\mathbb{R}}^{\mathcal{N}}$, $\mathcal{N}=\{1,...,N\}$, and the corresponding horofunction boundary. As a consequence, we describe the horofunctions for Hilbert's projective metric on the interior of the standard cone ${\mathbb{R}}_{+}^{\mathcal{N}}$ of ${\mathbb{R}}^{\mathcal{N}}$.

minor typos corrected