Centre for Computational and Discrete Geometry
University of Calgary
Convex 4-polytopes
Convex polytopes are fundamental objects that arise in many areas of pure and applied mathematics. Although, these objects are fairly well understood in dimensions 2 and 3, even in dimension 4 many important questions about the convex polytopes remain unanswered. One may readily speculate that a key reason beyond such a qualitative gap in understanding of polytopes is our inability to visualize in dimensions higher than three.
Within this project we investigate some gemoetric and combinatorial properties of convex 4-polytopes.
- Edge-antipodality
Let V be the set of vertices of a convex d-polytope P in real d-space, d ≥ 3. Two elements x and y are antipodal points of V, or antipodal vertices of P, if there exists distinct parallel supporting hyperplanes H(x) and H(y) of P such that x in H(x) and y in H(y).
We say that V or P is antipodal if any two elements of V are antipodal. Antipodal d-polytopes have been studied extensively over the past fifty years, and it is known that any such P has at most 2d vertices. In the last decade I. Talata introduced the concept of an edge-antipodal P: any two vertices of P, that lie on an edge of P, are antipodal. It is known that edge-antipodal 3-polytopes are antipodal, and that for each d ≥ 4, there is an edge-antipodal P that is not antipodal.
With K. Boroczky, we have began a
program for the study (classification, and determining the maximum of the number of vertices)
of edge – antipodal d-polytopes, d ≥ 4.
Most progress has been achieved in the case that d = 4. There the focus is on strongly
edge-antipodal P (any two vertices of P, that lie on the edge of P, are contained in
distinct parallel facets of P) and its polar.
- Separation
One of the most famous conjectures in Discrete Geometry is attributed to H. Hadwiger,
I.Z. Gohberg and A.S. Markus. It is that the smallest number h(K),
of smaller homothetic
copies of a compact convex set K in real d-space, d ≥ 2, with which it is possible
to cover K, is at most 2d only if K is
d-dimensional parallelotope.
The G-M-H conjecture is confirmed for d = 2, open for d ≥ 3, and has a rich history with various equivalent formulations. The formulation of particular interest is due to K. Bezdek: if the origin is in the interior of K then h(K) is the smallest number of hyperplanes required to strictly separate the origin from any face of the polar K* of K. This formulation is particularly attractive in the case of polytopes and leads naturally to the following Separation problem: let P be a convex d-polytope. Determine the smallest number s(P) such that any facet of P is strictly separated from an arbitrary fixed interior point of P by one of s(P) hyperplanes.
Again, most progress on the separation problem has been achieved in the case that d = 4 and the 4-polytope P is neighborly (any two vertices of P determine an edge of P). In particular, it is known that for certain classes of neighborly 4-polytopes P: s(P) ≤ 9. The big question now is if s(P) ≤ 9 for any neighborly 4-polytope?
- Combinatorics
The construction of new classes of convex polytopes is a fundamental area of research
in the study of the combinatorial properties of convex polytopes. In the past decade,
T. Bisztriczky introduced two new classes of
non-simplical convex d-polytopes P
that may be viewed as generalizations of cyclic d-polytopes: ordinary (2m+1)-polytopes
for m ≥1, and periodically cyclic Gale 2m-polytopes for m≥2. The introduction of these
polytopes lead to the discovery of multiplexes and multiplical polytopes, and with M. Bayer,
to the formulation of braxtopes and braxial polytopes.
The problem of interest is the realization of periodically cyclic Gale 2m-polytopes P(2m) in the case m =2. While there is a straightforward construction for m ≥ 3, the only known realization of periodically-cyclic Gale 4-polytopes are the bicyclic 4-polytopes introduced by Z. Smilansky. With J. Lawrence, we have classified these 4-polytopes, but it is not known if there are realizable 4-polytopes P(4) that are not bicyclic.
If you would like to learn more about this project, please contact Prof. Ted Bisztriczky.