The Horofunction Boundary of Finite Dimensional \(\ell_p\) Spaces
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 \(\mathbb{R}^n\), and the corresponding horofunction boundary. As a consequence, we describe the horofunctions for Hilbert’s projective metric on the interior of the standard cone \(\mathbb{R}_{+}^n\) of \(\mathbb{R}^n\).
Type
Publication
Colloquium Mathematicum, 155(1), 51–65
Status
Peer-reviewed
Open access
Main results
Metric functionals on \(\ell_1^n\)
Every metric functional on \(\ell_1^n\) is either internal or of the form
$$ h(x) = \sum_{i\in I}\epsilon_i x_i + \sum_{i\not\in I}(|x_i - \mu_i| - |\mu_i|),$$where \(\empty \neq I \subseteq [n]\), \((\epsilon_i) \in \{-1, 1\}^I\), and \((\mu_i) \in \mathbb{R}^{[n]\setminus I}\).
Metric functionals on \(\ell_p^n\) with \(1 < p < \infty\)
Every metric functional on \(\ell_p^n\) with \(1 < p < \infty\) is either internal or of the form
$$ h(x) = - \sum_{i\in [n]} \mu_i x_i\,,$$where \((\mu_i)\in \mathbb{R}^n\) with \(||\mu||_{q} = 1\) and \(q :=p(p-1)^{-1}\).
Authors
Research Scientist
I’m a Researcher in Mathematics. I’ve carried out research at Aalto University, Inria Saclay, and VTT. I’ve made contributions to metric geometry, functional analysis and optimization. I’m passionate about quantum computing, risk analysis and optimization algorithms.