AnR danalogue of Valentine’s theorem on 3-convex sets

This paper deals with anR ^danalogue of a theorem of Valentine which states that a closed 3-convex setS in the plane is decomposable into 3 or fewer closed convex sets. In Valentine’s proof, the points of local nonconvexity ofS are treated as vertices of a polygonP contained in the kernel ofS, yielding a decomposition ofS into 2 or 3 convex sets, depending on whetherP has an even or odd number of edges. Thus the decomposition actually depends onc(P′), the chromatic number of the polytopeP′ dual toP. A natural analogue of this result is the following theorem: LetS be a closed subset ofR ^d, and letQ denote the set of points of local nonconvexity ofS. We require thatQ be contained in the kernel ofS and thatQ coincide with the set of points in the union of all the (d − 2)-dimensional faces of somed-dimensional polytopeP. ThenS is decomposable intoc(P′) closed convex sets.