Abstract
A vertex v of a convex polygon P is called minimal (respectively maximal) if the circle going through v and its neighbouring vertices encloses the interior of P (respectively has no vertex of P in its interior) The main result of this paper is a generalization to the convex polytopes of R d of the following theorem: Every convex polygon has at least two minimal and two maximal vertices The proof uses a duality theory which translates some spherical properties of a convex polytope of R d into combinatorial properties of a convex polyhedron of R d+1.
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Schatteman, A. A four-vertex theorem for polygons and its generalization to polytopes. Geom Dedicata 34, 229–242 (1990). https://doi.org/10.1007/BF00181686
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DOI: https://doi.org/10.1007/BF00181686