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

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.