
Tibor
Bisztriczky Faculty
Professor and Professor Emeritus B.Sc., 1970, McMaster University Main Menu

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 edge – antipodal
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: s(P) ≤ 16. The big question now is if s(P) ≤ 16 for any neighborly 4polytope?
Topics in Combinatorial Geometry: ErdösSzekeres type theorems, Transversal properties of families of ovals in the
plane, and triangulations of simple convex polygons.
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Geometry Fest ,
Renyi Institute,June 1115,
2007, Budapest.
Canadian Mathematical
Bulletin, 52 (3), September 2009.
Associate
Member, Renyi Institue of
Mathematics, 201114
Adjunct Professor, DepT. Mathematics and Statistics/York University, 201218.
Faculty Professor and
Professor Emeritus,University
of Calgary
CONFERENCE
PHOTOS
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: CONVEX SETS AND APPLICATIONS, 2006
CALGARY : INTUITIVE GEOMETRY, 2007
CALGARY,
CONVEX AND ABSTRACT POLYTOPES, 2005
PITTSBURGH: AMS, CONVEXITY AND COMBINATORICS, 2004
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.