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The Geometry of L 0

  • N. J. Kalton (a1), A. Koldobsky (a1), V. Yaskin (a2) and M. Yaskina (a2)
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 Lp -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 Lp -spaces.

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References
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Canadian Journal of Mathematics
  • ISSN: 0008-414X
  • EISSN: 1496-4279
  • URL: /core/journals/canadian-journal-of-mathematics
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