This is an announcement for the paper "The geometry of $L_0$" by N.J.Kalton, A.Koldobsky, V.Yaskin and M.Yaskina. Abstract: Suppose that we have the unit Euclidean ball in $\R^n$ and construct new bodies using three operations - linear transformations, closure in the radial metric and multiplicative summation defined by $\|x\|_{K+_0L} = \sqrt{\|x\|_K\|x\|_L}.$ We prove that in dimension 3 this procedure gives all origin symmetric convex bodies, while this is no longer true in dimensions 4 and higher. We introduce the concept of embedding of a normed space in $L_0$ that naturally extends the corresponding properties of $L_p$-spaces with $p\ne0$, and show that the procedure described above gives exactly the unit balls of subspaces of $L_0$ in every dimension. We provide Fourier analytic and geometric characterizations of spaces embedding in $L_0$, and prove several facts confirming the place of $L_0$ in the scale of $L_p$-spaces. Archive classification: Functional Analysis; Metric Geometry Mathematics Subject Classification: 46B20, 52Axx Remarks: 21 pages The source file(s), lzero.tex: 51885 bytes, is(are) stored in gzipped form as 0412371.gz with size 15kb. The corresponding postcript file has gzipped size 80kb. Submitted from: yaskinv@math.missouri.edu The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/math.FA/0412371 or http://arXiv.org/abs/math.FA/0412371 or by email in unzipped form by transmitting an empty message with subject line uget 0412371 or in gzipped form by using subject line get 0412371 to: math@arXiv.org.