The Geometry of ${{L}_{0}}$

Kalton, N. J. ; Koldobsky, A. ; Yaskin, V. ; Yaskina, M.

Canadian journal of mathematics, Tome 59 (2007) no. 5, pp. 1029-1049

Résumé

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 ${{\left\| x \right\|}_{K+0L}}\,=\,\sqrt{{{\left\| x \right\|}_{K}}{{\left\| x \right\|}_{L}}}$ . 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.

Détail

DOI : 10.4153/CJM-2007-044-0

Mots-clés : 52A20, 52A21, 46B20

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