Curvature extrema and four-vertex theorems for polygons and polyhedra
Abstract
Discrete analogs of extrema of curvature and generalizations of the four-vertex theorem to the case of polygons and polyhedra are suggested and developed. 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 evolutes is obtained. Also are considered higher-dimensional analogs of the four-vertex theorem for regular and shellable triangulations.
Keywords
Cite
@article{arxiv.0805.0629,
title = {Curvature extrema and four-vertex theorems for polygons and polyhedra},
author = {Oleg R. Musin},
journal= {arXiv preprint arXiv:0805.0629},
year = {2010}
}
Comments
Several changes in the last section. In the original version of this paper we claimed that any regular triangulation of a convex d-polytope has at least d ears. For a proof we used the same arguments as in Schatteman's paper [22]. Since this paper has certain gaps (see our paper [1]), the d -ears problem of a regular triangulation is still open