
Tibor
Bisztriczky Faculty
Professor and Professor Emeritus B.Sc., 1970, McMaster University Associate Member, Renyi Institute, Hungarian Academy of Sciences,2011 Adjunct Professor of Mathematics & Statistics, York University,20122018 


Current research is in
the fields of convex polytopes, and discrete geometry. Specific topics of
interest include:
Edgeantipodality (A set V or, a polytope P with vertex set V, is antipodal if any two elements of V are antipodal.
Antipodal dpolytopes have been studied extensively over the past fifty years,
and it is known that any such P has at most 2^{d} vertices. In the last
decade I. Talata introduced the concept of an edgeantipodal P: any two vertices of P, that lie on an edge of P, are antipodal. It is known that
edgeantipodal 3polytopes are antipodal, and that for each d ≥ 4, there
is an edgeantipodal 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
edgeantipodal dpolytopes, d ≥ 4. Most progress has been achieved in the
case that d = 4. There the focus is presently on strongly
edgeantipodal
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 dspace, d ≥ 2, with which it is possible to cover K, is at most 2^{d}
only if K is ddimensional parallelotope. The GMH 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 dpolytope.
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 4polytope P is neighborly (any two vertices of P determine an edge of P).
In particular, it is known that for certain classes of neighborly 4polytopes P
that s(P) ≤ 16. The big question now is if s(P) ≤ 16 for any neighborly 4polytope?
Topics in Combinatorial Geometry: ErdosSzekeres type theorems, Transversal properties of families of ovals in the
plane, Edgeantipodal 4polytopes and Triangulations of simple convex polygons.
*****************************************************************************
CONFERENCE PHOTOS
19882017
BERLIN, CONVEX & DISCRETE
GEOMETRY, JOERGSHOP, JUNE 2017
ATLANTA, AMS, DISCRETE GEOMETRY & CONVEXITY
SPECIAL SESSION, JANUARY 2017
OAXACA, CMOBIRS, GEOMETRY,
COMBINATORICS&TOPOLOGY, OCTOBER 2016
BUDAPEST, DISCRETE GEOMETRY DAYS, JUNE 2016
BANFF (BIRS), DISCRETE GEOMETRY & SYMMETRY,
FEB. 2015
HALIFAX, CMS, DISCRETE &
COMBINATORIAL GEOMETRY SPECIAL SESSION, JUNE 2013
BANFF (BIRS), TRANSVERSAL THEORY
& HELLY TYPE THEOREMS, OCT. 2012
SZEGED, DISCRETE & CONVEX
GEOMETRY CONFERENCE, MAY 2012
RICHMOND, AMS, CONVEXITY &
COMBINATORICS SPECIAL SESSION, NOV. 2010
KELOWNA, PRAIRIE DISCRETE MATH
WORKSHOP, AUGUST 2009
SEVILLA, PHENOMINA IN HIGH
DIMENSIONS CONFERENCE, JUNE 2008
BANFF (BIRS) & CALGARY,
INTUITIVE GEOMETRY, SEPTEMBER 2007
BUDAPEST, GEOMETRY FEST, JUNE 2007
BANFF (BIRS), CONVEX GEOMETRY &
APPLICATIONS, MARCH 2006
SAN ANTONIO, AMS, CONVEX &
DISCRETE GEOMETRY SPECIAL SESSION, JANUARY 2006
BANFF (BIRS) & CALGARY, CONVEX
AND ABSTRACT POLYOTPES, MAY 2005
PITTSBURGH, AMS, CONVEXITY &
COMBINATORICS SPECIAL SESSION, NOVEMBER 2004
GAETA, COMBINATORICS CONFERENCE,
MAYJUNE 2000
HALIFAX, AFFFINE GEOMETRY
CONFERENCE, MAY 1996
MSRI, CONVEX GEOMETRY& GEOMETRIC
FUNCTIONAL ANALYSIS, MAY 1996
CATANIA, COMBINATORICS CONFERENCE,
SEPTEMBER 1991
VIENNA, MATHEMATISCHE KOLLOQIUM,
TECHNISCHE UNIVERSITAET, APRIL 1991
RAVELLO, COMBINATORICS CONFERENCE,
MAY 1988