  The 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.
