A geometric solution to a maximin problem involving determinants of sets of unit vectors in finite dimensional real or complex vector spaces

Abstract

Given $n+1$ unit vectors in $\mathbf{R}^n$ or $\mathbf{C}^n,$ consider the absolute values of the determinants of the vectors taken $n$ at a time. By taking a geometric perspective, we show that the minimum of these determinants is maximized when the vectors point from the origin to the vertices of a regular simplex inscribed in the unit sphere in $\mathbf{R}^n,$ even in the complex case. We also discuss variations on this problem and a few connections to other problems.