Tibor Bisztriczky’s Homepage


Tibor Bisztriczky

Faculty Professor and Professor Emeritus
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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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-polytopes  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



                                      CONFERENCE    PHOTOS


   Atlanta, ams, discrete geometry &convexity, January 2017


                                OAXACA, birs-cmo, transversal theorems, October 2016


                                  BUDAPEST, discrete geometry DAYS, june 2016      


                               Birs;   discrete geometry and symmetry, 2015 



                                 HALIFAX, DISCRETE GEOMETRY, JUNE 2013  



                              BIrs:  TRANSVERSALS AND HELLY TYPE THEOREMS, 2012



                                   Szeged, discrete & convex geometry, may 2012



                         RICHMOND, AMS, CONVEXITY  AND COMBINATORICS, 2010


                        Kelowna, prairie discrete math workshop, august 2009


                            Seville, phenomena in high dimensions, june 2008


                        birs &Calgary, Intuitive geometry , September 2007


                                          Budapest , geometry  fest, june 2007




                             Birs, convex geometry and applications, march 2006


                          SAN ANTONIO: CONVEX AND DISCRETE GEOMETRY, 2006



                         Birs &  CALGARY, CONVEX AND ABSTRACT POLYTOPES, 2005




                           PITTSBURGH:  AMS,  CONVEXITY AND COMBINATORICS, 2004 



                                                                                  GAETA, COMBINATORICS 2000



                                                HALIFAX, AFFINE  Geometry CONFERENCE , 1996


                                                                                  CATANIA, COMBINATORICS , september1991 

                               Vienna,  mathematische colloquium,april 1991



                                                                             Ravello, combinatorics , may 1988


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© 2002 D.B.