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.