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The usual theory of negative type (and $p$-negative type) is heavilydependent on an embedding result of Schoenberg, which states that a metricspace isometrically embeds in some Hilbert space if and only if it has2-negative type. A generalisation of this embedding result to the setting ofbi-lipschitz embeddings was given by Linial, London and Rabinovich. In thisarticle we use this newer embedding result to define the concept of distortedp-negative type and extend much of the known theory of p-negative type to thesetting 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 embeddinginto some Hilbert space with distortion at most $C$. Analogues of strict$p$-negative type and polygonal equalities in this new setting are given andsystematically studied. Finally, we provide explicit examples of these conceptsin the bi-lipschitz setting for the bipartite graphs $K_{m,n}$ and the Hammingcube $H_n$.

Negative type and bi-lipschitz embeddings into Hilbert space