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The geometry of $L_0$

N. J. Kalton·A. Koldobsky·V. Yaskin·M. Yaskina
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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 $\|x\|_{K+_0L} = \sqrt{\|x\|_K\|x\|_L}.$ 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.

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N. J. Kalton, A. Koldobsky, V. Yaskin, M. Yaskina. 2004-12-19. The geometry of $L_0$. https://arxiv.org/abs/math/0412371

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