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# Auerbach Bases and Minimal Volume Sufficient Enlargements

Published:2011-03-14
Printed: Dec 2011
• M. I. Ostrovskii,
Department of Mathematics and Computer Science, St. John's University, Queens, NY 11439, U.S.A.
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## Abstract

Let $B_Y$ denote the unit ball of a normed linear space $Y$. A symmetric, bounded, closed, convex set $A$ in a finite dimensional normed linear space $X$ is called a sufficient enlargement for $X$ if, for an arbitrary isometric embedding of $X$ into a Banach space $Y$, there exists a linear projection $P\colon Y\to X$ such that $P(B_Y)\subset A$. Each finite dimensional normed space has a minimal-volume sufficient enlargement that is a parallelepiped; some spaces have exotic'' minimal-volume sufficient enlargements. The main result of the paper is a characterization of spaces having exotic'' minimal-volume sufficient enlargements in terms of Auerbach bases.
 Keywords: Banach space, Auerbach basis, sufficient enlargement
 MSC Classifications: 46B07 - Local theory of Banach spaces 52A21 - Finite-dimensional Banach spaces (including special norms, zonoids, etc.) [See also 46Bxx] 46B15 - Summability and bases [See also 46A35]

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