Conditionally strictly negative definite kernels

TL;DR

This paper refines the concept of conditionally negative definite kernels to a stricter form, revealing surprising rigidity properties and characterizations, especially in relation to Coxeter groups and finite sets.

Contribution

It introduces the notion of conditionally strictly negative definite kernels and explores their properties, including characterizations on Coxeter groups and finite sets.

Findings

Word metric on Coxeter groups is conditionally strictly negative definite iff the group is a free product of Z2's.

Class of conditionally strictly negative definite kernels on finite sets is a one-parameter perturbation of strictly positive definite kernels.

Several examples illustrating the properties of these kernels.

Abstract

In this note we refine the notion of conditionally negative definite kernels to the notion of conditionally strictly negative definite kernels and study its properties. We show that the class of these kernels carries some surprising rigidity, in particular, the word metric function on Coxeter groups is conditionally strictly negative definite if and only if the group is a free product of a number of copies of $\mathbb{Z}_2$'s and that the class of conditionally strictly negative definite kernels on a finite set is a one-parameter perturbation of the class of strictly positive definite kernels on this set. We also discuss several examples.