MATLAB Examples

Verification of the Upper Bound Theorem (McMullen, 1970) in 4 dimensions

Contents

Introduction

The cyclic polytope $CP(n,d)$ may be defined as the convex hull of $n$ vertices on the moment curve $(t, t^2, t^3, ..., t^d)$. The precise choice of which $n$ points on this curve are selected is irrelevant for the combinatorial structure of this polytope. The number of $(d-1)$ -dimensional facets (faces) of $CP(n,d)$ is given by the formula:

$$f_i(\Delta(n,d)) = \frac{n}{n - \frac{d}{2}} {{n - \frac{d}{2}} \choose
n-d} \hspace{0.2in} d \quad \textrm{even} \quad $$

$$f_i(\Delta(n,d)) = 2 {{n - \frac{d+1}{2}} \choose n-d} \hspace{0.2in} d
\quad \textrm{odd} \quad $$

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.

Initial data

Set the points the convex hull of which is to be calculated.

t=(0:0.1:2)';
points=[t,t.^2,t.^3,t.^4];

Processing

Find the point identities defining each facet of the convex hull of the point set.

chull1=convhull_nd(points);

Post-processing

Find the number of the facets of the convex hull.

nfacets1=size(chull1,1)
nfacets1 =

   189

Calculate the number of points the convex hull of which was calculated, as well as their dimension.

[n,d]=size(points)
n =

    21


d =

     4

Calculate the maximum number of faces (3-dimensional facets) that a 4-dimensional convex hull ($$d=4$) of 21 points can have.

maxfacets=n/(n-d/2)*nchoosek(n-d/2,n-d)
maxfacets =

  189.0000

Check the validity of the Upper Bound Theorem.

if maxfacets<nfacets1
    error('Upper Bound Theorem not satisfied')
end

Contact author

(c) 2014 by George Papazafeiropoulos
First Lieutenant, Infrastructure Engineer, Hellenic Air Force
Civil Engineer, M.Sc., Ph.D. candidate, NTUA

Email: gpapazafeiropoulos@yahoo.gr

Website: http://users.ntua.gr/gpapazaf/