Auerbach bases and minimal volume sufficient enlargements

TL;DR

This paper characterizes finite-dimensional normed spaces with 'exotic' minimal-volume sufficient enlargements using Auerbach bases, revealing new geometric insights into the structure of these spaces.

Contribution

It provides a characterization of spaces with exotic minimal-volume sufficient enlargements based on Auerbach bases, advancing understanding of geometric properties of normed spaces.

Findings

Spaces with exotic minimal-volume sufficient enlargements are characterized by Auerbach bases.

Every finite-dimensional normed space has a minimal-volume sufficient enlargement that is a parallelepiped.

Some spaces possess non-standard, 'exotic' minimal-volume sufficient enlargements.

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 {\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 $P(B_Y)\subset A$. Each finite dimensional normed space has a minimal-volume sufficient enlargement which 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.