2004/07/14 by Stanislaw J. Szarek, Nicole Tomczak-Jaegermann
Mathematics · #math.FA #math.PR #msc:46B20 #msc:46B07 #msc:52A21 #msc:52A22 #msc:60D05
published as J. Funct. Anal. 221 (2005), no. 2, 407-438. · 35 p., LATEX; the paper is a follow up on math.FA/0407233
arxiv created 2004/07/14 · arxiv updated 2009/12/01
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 V. Milman. The proofs are probabilistic and depend on careful analysis of images of convex sets under Gaussian linear maps.