CURVATURE EXTREMES AND CAUSTICS OF POLYGONS AND POLYHEDRA

Oleg R. Musin

Abstract:

For a plane smooth curve $\gamma$ the cusps of its caustic $C(\gamma)$
correspond to the curvature extremes of $\gamma$.  Let $N_+$ and $N_-$ be the
numbers of positive and negative vertices of $\gamma$ and $ind(\gamma)$
denote its index. Then

$$N_+ - N_- = 2ind(\gamma) - 2ind(C(\gamma)).$$

The four--vertex theorem can be used to test the definition of a
global extreme of the curvature in discrete case. There are several discrete
versions of the four--vertex theorem. V.D. Sedykh proved that if a polygonal
line in ${\bf R}^3$ lies on the boundary of its convex hull then it has at
least four support vertices. In this paper we prove a $d$--dimensional version
of Sedykh's theorem: if a $d-1$--polyhedron in ${\bf R}^{d+1}$ lies on the
boundary of its convex hull then it has at least $2d$ supporting $d-2$--faces.
This theorem generalizes Schatteman's theorem for convex polyhedra.