Theorem on four support vertices of a polygonal line (Q1357916)

The author gives some nice discrete analogs to the four-vertex theorem for plane curves. For example, a closed non-self-intersecting convex space polygonal line \(P_n\) with \(n>3\) edges has at least four support vertices. A vertex \(v\) of \(P_n\) is a support vertex if there is a plane of support for \(P_n\) passing through \(v\) and the two neighboring vertices, and \(P_n\) is said to be convex if \(P_n\) belongs to the boundary of the convex hull of \(P_n\). Or, every non-self-intersecting regular plane \(n\)-gon \((n>3)\) has at least four support vertices. For plane closed polygonal lines support vertices are defined by support circles. See also the survey by \textit{B. Wegner} [Math. Pannon. 6, No. 1, 121-132 (1995; Zbl 0736.51016)].