Isometric embeddings of compact spaces into Banach spaces

We show the existence of a compact metric space K such that whenever K embeds isometrically into a Banach space Y , then any separable Banach space is linearly isometric to a subspace of Y . We also address the following related question: if a Banach space Y contains an isometric copy of the unit ball or of some special compact subset of a separable Banach space X, does it necessarily contain a subspace isometric to X? We answer positively this question when X is a polyhedral finite-dimensional space, c0 or 1. © 2008 Elsevier Inc. All rights reserved.