$\newcommand{\R}{\mathbb{R}}\newcommand{\D}[1]{\Delta_{#1}}\newcommand{\set}[1]{\{#1\}}\newcommand{\abs}[1]{\lvert#1\rvert}\newcommand{\E}{\mathbb{1}}$ In 1984 S.D.Berman, a Soviet mathematician, asked the following minmax question. Let $X$ be a $n \times 2$ real valued matrix. Let $x_i$ denote the $i$-th row of $X$, and let $\D{ij} = \det(x_i,x_j)$ denote the $ij$-minor of $X$. Define \begin{equation} \label{def:E} E(X) \equiv \frac{\max_{i<j}\abs{\D{ij}}}{\min_{i<j}\abs{\D{ij}}} \end{equation} Find $X$ that minimizes $E$.
My first question states the conjectured solution, Jeopardy-style. Let $\set{c_i \in \R^2}_{i=1}^{2n}$ be the vertices of a regular $2n$-polygon, centered at $0$, numbered counterclockwise. Let $C \equiv [c_1, \dots, c_n]$ be a row matrix comprised of the first $n$ vertices.
Is it true that $C$ minimizes $E(X)$?
I can prove it for $n=4,5$ using nothing but the Plucker equations. In what follows I ask two specialized questions. An affirmative answer to any of them implies the above conjecture for odd $n$.
Let $n=2d+1$ be an odd integer, and let $\D{} \equiv (\D{1}, \dots, \D{n}) \in \R^n$, $\D{s}>1$. Define \begin{equation} R_s(\D{}) = \D{s-1}^{-1} \begin{bmatrix} 1 & \D{s-1} \\ -\D{s} & 0 \end{bmatrix} \end{equation} where $\D{0} \equiv \D{n}$.
Consider a matrix equation \begin{equation} \tag{GP}\label{GP} \prod_{s=1}^n R_s(\D{}) = (-1)^d \E \end{equation} where $\E$ is the identity matrix.
Consider the extremal problem of minimizing $E_1(\D{}) \equiv \max_{s}\D{s}$ over $\R^n$ subject to the constraint \eqref{GP} and $\D{s}>1$. Let $\D{}^* \in \R^n$ be an extremal point.
Is it true that $\D{}^*$ is a constant vector, i.e. $\D{s}^* = \delta$ for some $\delta$?
This question is related to Chebyshev-like Problem for Plucker Coordinates that I asked earlier.
COMMENT. It might appear that \eqref{GP} imposes four scalar equations however only three of them are independent. Indeed, $\det{R_s} = \D{s} \D{s-1}^{-1}$ and \begin{equation} \det \prod_{s=1}^n R_s = \prod_{s=1}^n \D{s} \D{s-1}^{-1} = 1 \end{equation} One can prove that if $(\D{1}, \dots, \D{n})$ satisfies \eqref{GP} then so does $(\D{n}, \D{1}, \dots, \D{n-1})$ (cyclical symmetry) and $(\D{n}, \dots, \D{1})$ (mirror symmetry).
For my last question consider the locus of \eqref{GP} as an $n-3$ surface in $\R^n$.
Is it true that any $n-3$ sub-coordinates of $(\D{1}, \dots, \D{n})$ can be used to parametrize the surface (or a connectivity component if the surface is not connected)?
