
Professor
Tibor Bisztriczky 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?
Honours and Appointments:
Geometry Fest ,
Renyi Institute,June 1115,
2007, Budapest.
Canadian Mathematical
Bulletin, 52 (3), September 2009.
Associate
Member, Renyi Institue of
Mathematics, 201114.
Adjunct Professor,
Department of Mathematics and Statistics, York University, 20122015.
CONFERENCE PHOTOS
HALIFAX,
DISCRETE GEOMETRY, JUNE 2013
BIRS:
CONVEX SETS
AND APPLICATIONS, 2006