Projective geometry of polygons and discrete 4-vertex and 6-vertex theorems

1999/09/24 by V. Ovsienko, Ovsienko, V., S. Tabachnikov +1
Mathematics · #Differential Geometry (math.DG) #FOS: Mathematics #math.DG

paper · pdf · doi:10.48550/arxiv.math/9909150

LaTeX document, 16 pages, 5 figures

arxiv created 1999/09/24 · arxiv updated 2009/11/30

Abstract

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.

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