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

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