Tibor Bisztriczky’s Homepage
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Professor
Tibor Bisztriczky B.Sc., 1970, McMaster University Main Menu
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Research
Interests
Current research is in the fields of convex polytopes, and
discrete geometry . Specific topics of interest include:
- development of a class of convex d-polytopes whose
facial structure is determined by a total ordering of vertices;
- a study of the geometric and combinatorial properties
of convex 4-polytopes.The specific properties of
interest are the following :
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?
- the study of transversal properties of
families of congruent disks in the plane.
Honours and Appointments:
Geometry Fest , Renyi Institute,June 11-15, 2007, Budapest.
Canadian Mathematical Bulletin, 52 (3), September 2009.
Associate Member,
Renyi Institue of Mathematics, 2011-14.
Adjunct Professor, Department of Mathematics
and Statistics, York University, 2012-2015.
CONFERENCE PHOTOS
BIRS: CONVEX SETS AND
APPLICATIONS, 2006