Convex Hulls of Spatial Polygons with a Fixed Convex Projection

Let F be a convex n-gon in a horizontal plane of the Euclidean 3-space. Consider its spatial variation under which its vertices move vertically and let F be the convex hull of such a variation. In the general position, the boundary of F splits naturally into the "bottom" F 0 and the "top" F 00. The polyhedron F 0 (F 00) has triangular faces and no vertices inside. The projection of these faces on F yields a triangulation T 0 (T 00) of F. Obviously T 0 and (T 00) have no common diagonals. Suppose now that T 0 and (T 00) with no common diagonals are prescribed. Guibas conjectured that the appropriate variation exists. The present paper gives a suucient and a necessary condition of such existence. The necessary condition can fail which disproves the Guibas conjecture. MSC91: 52B10 A segment connecting two vertices of a at convex polygon and crossing its interior will be called a diagonal. Let F be a convex n{gon with no 3 vertices collinear. By a diagonal triangulation T of F, we will mean a nonempty set of its diagonals which do not intersect in intF and split F into triangles. One can see easily that these triangles are nondegenerate and their number is n ? 2, where n is necessarily 4. We will call them triangles of T.