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

Abstract

We give a complete description of the horofunction boundary of finite-dimensional ℓ p spaces, for 1 ≤ p ≤ ∞. We also study the variation norm on R N , 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 R N + of R N .

On the Metric Compactification of Infinite-dimensional Spaces

“…Although it is perhaps more often known as the horofunction compactification. For finite-dimensional Banach spaces, this compactification has been studied in [5,11,18,10,9].…”

“…In [9] the author gives a complete description of the metric compactification of ℓ p ({1, ..., N }) for all 1 ≤ p ≤ ∞ and N ∈ N. In this paper we study the metric compactification of ℓ p (J) for all 1 ≤ p < ∞ and J any countably infinite or uncountable index set.…”

Deep limits and cut-off phenomena for neural networks

“…This discussion can be compared with a similar one in Gutiérrez (2019). We summarize everything in the following proposition: Proposition 9 The metric functionals of the either half of the (or the full metric) Thompson metric that arise as limits (also called horofunctions) are given as follows: For d 1 we get that there exists a non-zero function u ∶ I → [0, 1] such that…”

Firm non-expansive mappings in weak metric spaces

“…Metric functionals and the related notion of horofunctions are discussed in [8,7,15,16]. These objects have been studied in spaces such as Hilbert and Thompson geometries [25,18,26], Teichmüller geometry [27], and normed spaces [24,12,9,8,10].…”