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.

MSC Classifications:

52A20, 52A21, 46B20 show english descriptions Convex sets in $n$ dimensions (including convex hypersurfaces) [See also 53A07, 53C45]
Finite-dimensional Banach spaces (including special norms, zonoids, etc.) [See also 46Bxx]
Geometry and structure of normed linear spaces 52A20 - Convex sets in $n$ dimensions (including convex hypersurfaces) [See also 53A07, 53C45]
52A21 - Finite-dimensional Banach spaces (including special norms, zonoids, etc.) [See also 46Bxx]
46B20 - Geometry and structure of normed linear spaces

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