  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.
