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Sufficient enlargements of minimal volume for two-dimensional normed spaces

Published online by Cambridge University Press:  07 September 2004

M. I. OSTROVSKII
M. I. OSTROVSKII
Affiliation:
Department of Mathematics, The Catholic University of America, Washington, D.C. 20064, U.S.A. e-mail: ostrovskii@cua.edu
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Abstract

We denote by $B_Y$ the unit ball of a normed linear space $Y$.

Definition. A symmetric, bounded, closed, convex set $A$ in a finite dimensional normed 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 $P{:}\,Y\,{\to}\,X$ such that $P(B_Y)\,{\subset}\,A$.

Minimal-volume sufficient enlargements are determined for two-dimensional spaces. The main results are:

(i) Each minimal-volume sufficient enlargement for a two-dimensional space is a parallelogram or a hexagon.

(ii) If a two-dimensional normed space $X$ has a minimal-volume sufficient enlargement that is not a parallelogram, then $B_X$ is linearly equivalent to the regular hexagon.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 137 , Issue 2 , September 2004 , pp. 377 - 396
Copyright
2004 Cambridge Philosophical Society

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