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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-polytope  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

             

           OAXACA, birs-cmo, transversal theorems, October 2016

                   dgd13   

                       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.jpg 

 

                                     BIRS:  CONVEX  SETS AND APPLICATIONS,   2006

 

 

                                                               CALGARY :  INTUITIVE GEOMETRY,  2007

 

                

                             CALGARY, CONVEX AND ABSTRACT POLYTOPES, 2005

 

                                       pittsburgh.jpg

 

                           PITTSBURGH:  AMS,  CONVEXITY AND COMBINATORICS, 2004 

 

                                             

 

                                    SAN ANTONIO: CONVEX AND DISCRETE GEOMETRY, 2006

 


                                                      
braxy

 

                                                                  BUDAPEST:  GEOMETRY FEST, 2007

 

                                                       

 

                                                                                  GAETA, COMBINATORICS 2000

                                                     

                                                HALIFAX, AFFINE  Geometry CONFERENCE , 1996

                                

                                                                                  CATANIA, COMBINATORICS 1991    

                                                                               

                                                                                             Ravello, combinatorics 1988

 

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© 2002 D.B.