On near polygons all whose hexes are dual polar spaces (Q2191261)

The aim of this paper is to provide a (relatively) elementary proof of the theorem by \textit{A. E. Brouwer} and \textit{A. M. Cohen} [Geom. Dedicata 20, 181--199 (1986; Zbl 0585.51010)] stating that a finite near \(2d\)-gon \(\mathcal S\) with \(d\geq 3\) such that \begin{itemize} \item \(\mathcal S\) contains at least three points \item any two points in \(\mathcal S\) at distance \(2\) have at least three common neighbours \item for any hex \(H\) of \(\mathcal S\), the subgeometry \(\widetilde{H}\) defined on the point-set of \(H\) whose lines are the lines of \(\mathcal S\) fully contained in \(H\) is a dual polar space. \end{itemize} The proof presented here relies on geometrical and algebraic combinatorial techniques, as well as on the theory of distance-regular graphs, without making recourse to Tits' theory of buildings. As such it might be more accessible to people familiar with finite geometry.