Tight Embeddability of Proper and Stable Metric Spaces

We introduce the notions of almost Lipschitz embeddability and nearly isometric embeddability. We prove that for p is an element of [1, infinity], every proper subset of L-p is almost Lipschitzly embeddable into a Banach space X if and only if X contains uniformly the l(p)(n)'s. We also sharpen a result of N. Kalton by showing that every stable metric space is nearly isometrically embeddable in the class of reflexive Banach spaces.