Sufficient enlargements of minimal volume for finite dimensional normed linear spaces

Details Sufficient enlargements of minimal volume for finite dimensional normed linear spaces Sufficient enlargements of minimal volume for finite dimensional normed linear spaces

2008-11-11

arxiv.org/abs/0811.1701v1

J. Funct. Anal. 255 (2008), no. 3, 589-619

Mathematics

Functional Analysis

Scientific paper

10.1016/j.jfa.2008.04.012

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 {\it 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 $P(B_Y)\subset A$. The main results of the paper: {\bf (1)} Each minimal-volume sufficient enlargement is linearly equivalent to a zonotope spanned by multiples of columns of a totally unimodular matrix. {\bf (2)} If a finite dimensional normed linear space has a minimal-volume sufficient enlargement which is not a parallelepiped, then it contains a two-dimensional subspace whose unit ball is linearly equivalent to a regular hexagon.

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