Additive Combination Spaces

TL;DR

This paper introduces $p$-additive combination spaces, providing a way to analyze their $p$-negative type behavior through components, generalizing previous work on metric trees and applicable to semi-metric spaces.

Contribution

It presents a formula for the $p$-negative type gap of $p$-additive combination spaces based on their components, extending prior results to semi-metric spaces.

Findings

Derived a formula for the $p$-negative type gap in these spaces.

Applicable to semi-metric spaces without relying on the triangle inequality.

Generalized earlier work on metric trees by Doust and Weston.

Abstract

We introduce a class of metric spaces called $p$-additive combinations and show that for such spaces we may deduce information about their $p$-negative type behaviour by focusing on a relatively small collection of almost disjoint metric subspaces, which we call the components. In particular we deduce a formula for the $p$-negative type gap of the space in terms of the $p$-negative type gaps of the components, independent of how the components are arranged in the ambient space. This generalizes earlier work on metric trees by Doust and Weston. The results hold for semi-metric spaces as well, as the triangle inequality is not used.