The geometry of L0

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 parallel to x parallel to(K+0L)= root parallel to x parallel to K parallel to x parallel to 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 not equal 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.