We introduce the notion of a firm non-expansive mapping in weak metric spaces, extending previous work for Banach spaces and certain geodesic spaces. We prove that, for firm non-expansive mappings, the minimal displacement, the linear rate of escape, and the asymptotic step size are all equal. This generalises a theorem by Reich and Shafrir.
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 $\\mathbf{R}^\\mathcal{N}$.