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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.

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                        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.jpg

 

                           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.