Professor and Professor Emeritus
B.Sc., 1970, McMaster University
Current research is in the fields of convex polytopes, and discrete geometry . Specific topics of interest include:
Edge-antipodality (A set V or,a polytope P with vertex set V, 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 presently 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) .
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) ≤ 16. The big question now is if s(P) ≤ 16 for any neighborly 4-polytope?
Topics in Combinatorial Geometry: Erdös-Szekeres type theorems, Transversal properties of families of ovals in the plane, Edge –antipodal 4-polytope and Triangulations of simple convex polygons.
Associate Member, Renyi InstituTe of Mathematics, 2011-14
Adjunct Professor, DepT. Mathematics and Statistics/York University, 2012-18.
Faculty Professor and Professor Emeritus, University of Calgary
BUDAPEST, discrete geometry DAYS, june 2016
Birs; discrete geometry and symmetry, 2015
HALIFAX, DISCRETE GEOMETRY, JUNE 2013
BIrs: TRANSVERSALS AND HELLY TYPE THEOREMS, 2012
RICHMOND, AMS, CONVEXITY AND COMBINATORICS, 2010
BIRS: CONVEX SETS AND APPLICATIONS, 2006
CALGARY : INTUITIVE GEOMETRY, 2007
CALGARY, CONVEX AND ABSTRACT POLYTOPES, 2005
PITTSBURGH: AMS, CONVEXITY AND COMBINATORICS, 2004
SAN ANTONIO: CONVEX AND DISCRETE GEOMETRY, 2006
BUDAPEST: GEOMETRY FEST, 2007
GAETA, COMBINATORICS 2000
HALIFAX, AFFINE Geometry CONFERENCE , 1996
CATANIA, COMBINATORICS 1991
Ravello, combinatorics 1988
© 2002 D.B.