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

Published online by Cambridge University Press:  20 November 2018

N. J. Kalton ,
A. Koldobsky ,
V. Yaskin  and
M. Yaskina
N. J. Kalton
Affiliation:
Department of Mathematics, University of Missouri, Columbia, MO 65211, U.S.A. email: nigel@math.missouri.edu, koldobsk@math.missouri.edu
A. Koldobsky
Affiliation:
Department of Mathematics, University of Missouri, Columbia, MO 65211, U.S.A. email: nigel@math.missouri.edu, koldobsk@math.missouri.edu
V. Yaskin
Affiliation:
Department of Mathematics, University of Oklahoma, Norman, OK 73019, U.S.A. email: vyaskin@math.ou.edu, myaskina@math.ou.edu
M. Yaskina
Affiliation:
Department of Mathematics, University of Oklahoma, Norman, OK 73019, U.S.A. email: vyaskin@math.ou.edu, myaskina@math.ou.edu
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Abstract

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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.

Keywords

52A2052A2146B20
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 59 , Issue 5 , 01 October 2007 , pp. 1029 - 1049
Copyright
Copyright © Canadian Mathematical Society 2007

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