The Regular Polygon Minimizes The Ratio of Plucker Coordinates on The Positive Grassmannian

Abstract

For a point $x$ on the Positive Grassmannian of two-dimensional subspaces in $\mathbb{R}^n$, define the loss function $E(x)$ as the ratio of its largest and smallest Plucker coordinates. We solve the extremal problem of minimizing the loss function $E(x)$ over the Grassmannian. This minimax problem was posed by Berman, et al. in their paper on error-correcting codes over the real numbers.