THE MAYER-VIETORIS AND IC EQUATIONS FOR CONVEX POLYTOPES

In this paper we use geometric dissection to obtain linear equations o n the flag vectors on convex polytopes. These results provide new proo fs and expressions of the complete system of such equations originally discovered by Bayer and Billera. The Mayer-Vietoris equation applies to a situation where two convex polytopes overlap to produce union and intersection, both convex polytopes. The operators I and C applied to a polytope produce the cylinder (or prism) and cone (or pyramid), res pectively, with the given polytopes as base. The IC equation relates t he flag vectors of the polytopes obtained in this way. As a consequenc e, it becomes easier to define linear functions of the flag vector, vi a initial data and their law of transformation under the operators I a nd C.