This is an announcement for the paper "Saturating constructions for normed spaces II" by Stanislaw J. Szarek and Nicole Tomczak-Jaegermann. Abstract: 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. Archive classification: Functional Analysis; Probability Mathematics Subject Classification: 46B20; 46B07; 52A21; 52A22; 60D05 Remarks: 35 p., LATEX The source file(s), SzarekTomczakSat2.tex: 104176 bytes, is(are) stored in gzipped form as 0407234.gz with size 33kb. The corresponding postcript file has gzipped size 127kb. Submitted from: szarek@cwru.edu The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/math.FA/0407234 or http://arXiv.org/abs/math.FA/0407234 or by email in unzipped form by transmitting an empty message with subject line uget 0407234 or in gzipped form by using subject line get 0407234 to: math@arXiv.org.