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

TL;DR

This paper provides a geometric approach to maximize the minimum absolute determinant of any n vectors chosen from n+1 unit vectors in real or complex n-dimensional space, revealing the optimal configuration as a regular simplex.

Contribution

It introduces a geometric solution to a maximin determinant problem, establishing that the optimal vector configuration is a regular simplex inscribed in the unit sphere.

Findings

Maximum of the minimum determinants achieved by regular simplex vertices.

The result extends to both real and complex vector spaces.

Discusses related variations and connections to other mathematical problems.

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.