Spaces with the maximal projection constant revisited

TL;DR

This paper characterizes n-dimensional normed spaces with maximal projection constants, linking them to geometric structures like zonotopes and equiangular vectors, and finds uniqueness in the 2D real case.

Contribution

It provides a complete geometric characterization of spaces with maximal projection constants, connecting them to equiangular sets and zonotopes, and identifies the unique 2D real case.

Findings

Spaces with maximal projection constant are isometric to certain zonotope-related spaces.

In the real 2D case, the maximal projection constant space is uniquely the affine regular hexagon.

The characterization involves the dual unit ball being between convex hulls of equiangular vectors.

Abstract

Let $n \geq 2$ be an integer such that an equiangular set of vectors $w_1, \ldots, w_d$ of the maximal possible cardinality (in relation to the the general Gerzon upper bound) exists in $\mathbb{K}^n$, where $\mathbb{K}=\mathbb{R}$ or $\mathbb{K}=\mathbb{C}$ (i.e. $d=\frac{n(n+1)}{2}$ in the real and $d=n^2$ in the complex case). We provide a complete characterization of $n$-dimensional normed spaces $X$ having a maximal absolute projection constant among all $n$-dimensional normed spaces over $\mathbb{K}$. The characterization states that $X$ has a maximal projection constant if and only if it is isometric to a space, for which the unit ball of the dual space is contained between the absolutely convex hull of the vectors $w_1, \ldots, w_d$ and an appropriately rescaled zonotope generated by the same vectors. As a consequence, we obtain that in the considered situations, the case of $n=2$ and $\mathbb{K}=\mathbb{R}$ is the only one, where there is a unique norm in $\mathbb{K}^n$ (up to an isometry) with the maximal projection constant. In this case, the unit ball is an affine regular hexagon in $\mathbb{R}^2$.