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.
Categories: math.MG