Negative Type and Bi-lipschitz Embeddings into Hilbert Space

Abstract

AbstractThe 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 $$(X,d_{X})$$ ( X , d X ) has p-negative type with distortion C ($$0\le p<\infty $$ 0 ≤ p < ∞ , $$1\le C<\infty $$ 1 ≤ C < ∞ ) if and only if $$(X,d_{X}^{p/2})$$ ( X , d X p / 2 ) 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 $$K_{m,n}$$ K m , n .