A four-vertex theorem for polygons and its generalization to polytopes (Q915131)

\textit{H. Guggenheimer} [Geom. Dedicata 12, 371-381 (1982; Zbl 0491.52008)] proved that every smooth, convex, plane curve C, whose radius is bounded away from zero, has at least two distinct circles of curvature which are contained in the compact domain defined by C and also at least two distinct circles of curvature of which the convex hull contains C. The aim of this note is to prove a similar theorem for convex polygons and to extend this version to polytopes. For this purpose including and excluding traces T of vertices of a polytope P in \({\mathbb{R}}^ d\) are introduced which generate a sphere having all vertices of P in its convex hull resp. no vertex of P is locate in the interior of that convex hull. T is called a k-trace if the smallest sphere through T has dimension k-1. A side trace is a (d-1)- trace which is contained in a facet of P. A (d-1)-trace is called extreme if it is the intersection of two side-traces of the same d-trace. With these notions the author proves the following results: 1) Let P be a convex polytope in \({\mathbb{R}}^ d\), and let \(\{T_ 1,...,T_ r\}\) be the family of all its including d-traces. Then the family of polytopes \(P_ i=conv T_ i\) satisfies (a) \(P_ i\cap P_ j\) is a common face of \(P_ i\) and \(P_ j\) for all i, j, (b) \(\cup^{r}_{i=1}P_ i=P\). A second decomposition of this type is obtained for the family of excluding d-traces of P. 2) P possesses at least d extreme including (d-2)-traces and at least d extreme excluding (d-2)-traces. For convex polygons the last result is the announced four-vertex theorem. Other kinds of four-vertex theorems for polytopes have been obtained by \textit{S. Bilinski} [Period. Math.-Phys. Astron., II. Ser. 18, 85-93 (1963; Zbl 0121.376)] and the reviewer [On the evolutes of piecewise linear curves in the plane, Preprint TU Berlin (1990)].