
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, Edge –antipodal 4polytope and Triangulations of simple convex
polygons.
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Associate
Member, Renyi InstituTe of
Mathematics, 201114
Adjunct Professor, DepT. Mathematics and Statistics/York University, 201218.
Faculty Professor and
Professor Emeritus, University of Calgary
CONFERENCE
PHOTOS
OAXACA,
birscmo, 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
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
HALIFAX, AFFINE
Geometry CONFERENCE , 1996
CATANIA,
COMBINATORICS 1991
Ravello, combinatorics
1988
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© 2002 D.B.