2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/79150The paper concerns discrete versions of the three well-known results of projective differential geometry: the four vertex theorem, the six affine vertex theorem and the Ghys theorem on four zeroes of the Schwarzian derivative. We study geometry of closed polygonal lines in $\bbRP^d$ and prove that polygons satisfying a certain convexity condition have at least d+1 flattenings. This result provides a new approach to the above mentioned classical theorems.LaTeX document, 16 pages, 5 figuresDifferential Geometrytext