Auerbach Bases and Minimal Volume Sufficient Enlargements

http://dx.doi.org/10.4153/CMB-2011-043-3
Canad. Math. Bull. 54(2011), 726-738

  Published:2011-03-14
 Printed: Dec 2011

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.