Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: The geometry of $L_0$   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.

29
The categories of this paper are: funct-an
1
The categories of this paper are: cs
33
The categories of this paper are: plasm-ph
7
The categories of this paper are: hep-th
20
The categories of this paper are: acc-phys
8
The categories of this paper are: hep-ex
37
The categories of this paper are: supr-con
14
The categories of this paper are: math-ph
26
The categories of this paper are: bayes-an
12
The categories of this paper are: nucl-th
35
The categories of this paper are: mtrl-th
15
The categories of this paper are: nucl-ex
11
The categories of this paper are: hep-lat
22
The categories of this paper are: chao-dyn
23
The categories of this paper are: solv-int
5
The categories of this paper are: gr-qc