Sufficient enlargements of minimal volume for finite dimensional normed linear spaces

TL;DR

This paper characterizes minimal-volume sufficient enlargements in finite-dimensional normed spaces, showing they are related to zonotopes from totally unimodular matrices and identifying conditions involving regular hexagons.

Contribution

It provides a complete description of minimal-volume sufficient enlargements, linking them to zonotopes and geometric structures like regular hexagons in subspaces.

Findings

Minimal-volume sufficient enlargements are linearly equivalent to zonotopes from totally unimodular matrices.

Spaces with non-parallelepiped minimal enlargements contain 2D subspaces with regular hexagon unit balls.

Characterization of sufficient enlargements in terms of geometric and algebraic properties.

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 $P(B_Y)\subset A$. 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.