Verification of the Upper Bound Theorem (McMullen, 1970) in 4 dimensions
The cyclic polytope may be defined as the convex hull of vertices on the moment curve . The precise choice of which points on this curve are selected is irrelevant for the combinatorial structure of this polytope. The number of -dimensional facets (faces) of is given by the formula:
The upper bound theorem states that cyclic polytopes have the largest possible number of faces among all convex polytopes with a given dimension and number of vertices.
Set the points the convex hull of which is to be calculated.
Find the point identities defining each facet of the convex hull of the point set.
Find the number of the facets of the convex hull.
nfacets1 = 189
Calculate the number of points the convex hull of which was calculated, as well as their dimension.
n = 21 d = 4
Calculate the maximum number of faces (3-dimensional facets) that a 4-dimensional convex hull () of 21 points can have.
maxfacets = 189.0000
Check the validity of the Upper Bound Theorem.
if maxfacets<nfacets1 error('Upper Bound Theorem not satisfied') end
(c) 2014 by George Papazafeiropoulos First Lieutenant, Infrastructure Engineer, Hellenic Air Force Civil Engineer, M.Sc., Ph.D. candidate, NTUA