Negative type and bi-lipschitz embeddings into Hilbert space

The usual theory of negative type (and $p$-negative type) is heavily dependent on an embedding result of Schoenberg, which states that a metric space isometrically embeds in some Hilbert space if and only if it has 2-negative type. A generalisation of this embedding result to the setting of bi-lipschitz embeddings was given by Linial, London and Rabinovich. In this article we use this newer embedding result to define the concept of distorted p-negative type and extend much of the known theory of p-negative type to the setting of bi-lipschitz embeddings. In particular we show that a metric space $(X; d_X)$ has $p$-negative type with distortion $C$ $(0 \le p < \infty$, $1 \le C < 1$) if and only if $(X; d^{p/2}_X$) admits a bi-lipschitz embedding into some Hilbert space with distortion at most $C$. Analogues of strict $p$-negative type and polygonal equalities in this new setting are given and systematically studied. Finally, we provide explicit examples of these concepts in the bi-lipschitz setting for the bipartite graphs $K_{m,n}$ and the Hamming cube $H_n$.