horofunction compactification

We present a complete characterization of the metric compactification of $L_p$ spaces for $1 \\leq p

The notion of metric compactification was introduced by Gromov and later rediscovered by Rieffel. It has been mainly studied on proper geodesic metric spaces. We present here a generalization of the metric compactification that can be applied to infinite-dimensional Banach spaces. Thereafter we give a complete description of the metric compactification of infinite-dimensional $\\ell_p$ spaces for all $1 \\leq p

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 $\\mathbf{R}^\\mathcal{N}$.