Curvature extrema and four-vertex theorems for polygons and polyhedra

TL;DR

This paper develops discrete analogs of curvature extrema and four-vertex theorems for polygons and polyhedra, establishing formulas and higher-dimensional generalizations to extend classical differential geometry results.

Contribution

It introduces new discrete curvature extrema concepts and extends four-vertex theorems to polygons and polyhedra, including higher-dimensional analogs.

Findings

Derived a formula linking curvature extrema to winding numbers.

Extended four-vertex theorem to polygons and polyhedra.

Proposed higher-dimensional analogs for triangulations.

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.