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 vert...

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Bibliographic Details
Published in:arXiv.org 2016-08
Main Author: Fincher, Mark
Format: Article
Language:English
Subjects:
Apexes
Determinants
Vector spaces
Online Access:Get full text
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Summary: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.
ISSN:2331-8422