Conditionally strictly negative definite kernels

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.