The horofunction boundary of finite-dimensional $\ell_{p}$ spaces

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 $\mathbf{R}^\mathcal{N}$ , $\mathcal{N}=\{1,…,N\}$ , and the corresponding horofunction boundary. As a consequence, we describe the horofunctions for Hilbert’s projective metric on the interior of the standard cone $\mathbf{R}^\mathcal{N}_{+}$ of $\mathbf{R}^\mathcal{N}$ .