Abstract view
Published:2007-10-01
Printed: Oct 2007
N. J. Kalton
A. Koldobsky
V. Yaskin
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.