The geometry of $L_0$

Abstract

Suppose that we have the unit Euclidean ball in ${\mathbb{R}}^n$ and construct new bodies using three operations - linear transformations, closure in the radial metric and multiplicative summation defined by $\Vert x{\Vert }_{K{+}_{0}L}=\sqrt{\Vert x{\Vert }_K\Vert x{\Vert }_L}.$ 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 $p\ne 0$, 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.

21 pages