The horofunction boundary of finite-dimensional $\ell_p$ spaces

TL;DR

This paper provides a comprehensive description of the horofunction boundary for finite-dimensional ll_p spaces, the variation norm, and Hilbert's projective metric, enhancing understanding of their geometric boundaries.

Contribution

It offers the first complete characterization of the horofunction boundary for finite-dimensional ll_p spaces and related metrics, including the variation norm and Hilbert's projective metric.

Findings

Complete description of the horofunction boundary for ll_p spaces

Characterization of horofunctions for the variation norm

Description of horofunctions for Hilbert's projective metric

Abstract

We give a complete description of the horofunction boundary of finite-dimensional $\ell_p$ spaces for $1\leq p\leq \infty$. 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 $\mathbb{R}^{\mathcal{N}}$.