Saturating constructions for normed spaces II

We prove several results of the following type: given finite-dimensional normed space V possessing certain geometric property there exists another space X having the same property and such that (1) log dim X = O(log dim V ) and (2) every subspace of X, whose dimension is not “too small”, contains a further well-complemented subspace nearly isometric to V. This sheds new light on the structure of large subspaces or quotients of normed spaces (resp., large sections or linear images of convex bodies) and provides definitive solutions to several problems stated in the 1980s by Milman. © 2004 Elsevier Inc. All rights reserved.