Spaces with the maximal projection constant revisited

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 $\mathbb{R}^2$.