$\newcommand{\R}{\mathbb{R}}\newcommand{\D}[1]{\Delta_{#1}}\newcommand{\set}[1]{\{#1\}}\newcommand{\abs}[1]{\lvert#1\rvert}$ Let $n=2d+1$ be an odd integer, let $Gr(2,n)$ denote the Grassmmanian over $\R$. For $x \in Gr(2,n)$ let $\D{ij}(x)$ denote its Plucker coordinates.
Define two subsets of the index set $1 \leq i < j \leq n$: \begin{equation} N \equiv \set{ij : (j-i)=\pm 1 \mod n} \end{equation} and \begin{equation} F \equiv \set{ij : (j-i)=\pm d \mod n} \end{equation} Note that $\abs{N} = \abs{F} = n$. One can prove that $2n$-vector $(\D{ij}(x))$, $ij \in N \cup F$, uniquely identifies $x \in Gr(2,n)$
Consider the extremal problem of minimizing $E(x) \equiv \max_{ij \in F}\D{ij}(x)$ over $x \in Gr(2,n)$ subject to constraints $\D{ij}(x)=1, ij \in N$. Let $x_* \in Gr(2,n)$ be an extremal point.
Is it true that $\D{ij}(x_*) = \delta$, $ij \in F$, for some $\delta$?
I can prove it for $n=5$, but the proof doesn't generalize.
