This is an announcement for the paper "Tensor products of convex sets and the volume of separable states on N qudits" by Guillaume Aubrun and Stanislaw J. Szarek. Abstract: This note deals with estimating the volume of the set of separable mixed quantum states when the dimension of the state space grows to infinity. This has been studied recently for qubits; here we consider larger particles. We also show that the partial transpose criterion becomes weaker when the dimension increases, and that the lower bound $6^{-N/2}$ on the (Hilbert-Schmidt) inradius of the set of separable states on $N$ qubits obtained recently by Gurvits and Barnum is essentially optimal. We employ standard tools of classical convexity, high-dimensional probability and geometry of Banach spaces; one relatively new point is a formal introduction of the concept of projective tensor products of convex bodies, and an initial study of this concept. PACS numbers: 03.65.Ud,03.67.-a,03.65.Db,02.40.Ft,02.50.Cw MSC-class: 46B28, 47B10, 47L05, 52A38, 81P68 Archive classification: Quantum Physics; Functional Analysis Remarks: 14 pages The source file(s), , is(are) stored in gzipped form as with size . The corresponding postcript file has gzipped size . Submitted from: szarek@cwru.edu The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/quant-ph/0503221 or http://arXiv.org/abs/quant-ph/0503221 or by email in unzipped form by transmitting an empty message with subject line uget 0503221 or in gzipped form by using subject line get 0503221 to: math@arXiv.org.