The Geometry of L0

Abstract 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 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 ≠ 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.