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

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