Tibor Bisztriczky’s Homepage


Professor Tibor Bisztriczky
Rm# 422 Mathematical Sciences Building 
2500 University Drive N.W. Calgary, AB T2N 1N4
Phone: (403)220-6312

B.Sc., 1970, McMaster University
M.Sc., 1971, McMaster University
Ph.D., 1974, University of Toronto

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Research Interests


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?


Honours   and  Appointments:  


         Geometry Fest , Renyi Institute,June 11-15, 2007, Budapest.


                        Canadian Mathematical Bulletin, 52 (3), September 2009.


                      Associate Member, Renyi Institue of Mathematics, 2011-14.


                      Adjunct Professor, Department of Mathematics and Statistics, York University, 2012-2015.



                                                               CONFERENCE    PHOTOS   


                     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









                                                                         SAN ANTONIO: CONVEX AND DISCRETE GEOMETRY, 2006




                                                                              BUDAPEST:  GEOMETRY FEST, 2007



                                       GAETA,  COMBINATORICS 2000


                                              CATANIA, COMBINATORICS 1991    


                                                 Ravello, combinatorics 1988


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© 2002 D.B.