Curvature Extrema and Four-Vertex Theorems for Polygons and Polyhedra

Discrete analogs of extrema of curvature and generalizations of the four-vertex theorem to the case of polygons and polyhedra are suggested and developed. Several interrelated approaches are considered. For smooth curves and polygonal lines in the plane, a formula relating the number of extrema of curvature to the winding numbers of the curves (polygonal lines) and their caustics is obtained. One of the main results in the higher-dimensional case says that any regular triangulation of a d-ball containing at least d simplices has at least d so-called ears. Bibliography: 25 titles.