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

الأصل · EN

Given n+1 unit vectors in Rⁿ or Cⁿ, 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 Rⁿ, even in the complex case. We also discuss variations on this problem and a few connections to other problems.