Some semicontinuity theorems for convex polytopes and cell-complexes

In a finite-dimensional EUCLIDean space E , a convex polytope P is a set which is the convex hull of a finite set; here such a set P will simply be called a polytope, or, if it is d-dimensional, a d-polytope. 2) An s-/ace of P is an s-dimensional set (necessarily an s-polytope) which is either P itself or is the intersection of P with a supporting hyperplane. The number of s-faces of P will be denoted by [~(P) and the s-measure of their union by ~ ( P ) . Our section headings are as follows : 1. The functions /~ for cell-complexes 2. A pulling process 3. The functions $~ for cell-complexes 4. The functions $~ for convex polytopes 5. Inequalities for the functions $~ 6. Intersection properties of simplices 7. Sections of simplices Each polytope is a compact set, and thus the class of all polytopes in E can be metrized by the HAUSDORFF distance. Our principal aim in w 1 is to show that when the polytopes are topologized in this way, each of the functions /~ is lower semicontinuous; thus for each polytope P in E it is true t ha t all polytopes sufficiently close to P have at least as m a n y s-faces as P . In connection with various extremal problems, this result can be used to restrict a t tent ion to polytopes whose vertices are in general position, bu t for t h a t purpose the pulling process described in w 2 is more advantageous. Lower semicont inui ty of the functions $~ is established in w167 3) The reasoning of w 3, like t ha t of w 1, applies not only to convex polytopes bu t also