  We prove several results of the following type: given finite dimensional
normed space V there exists another space X with log (dim X) = O(log (dim V))
and such that every subspace (or quotient) of X, whose dimension is not "too
small," contains a further subspace isometric to V. This sheds new light on the
structure of such large subspaces or quotients (resp., large sections or
projections of convex bodies) and allows to solve several problems stated in
the 1980s by V. Milman. The proofs are probabilistic and depend on careful
analysis of images of convex sets under Gaussian linear maps.
