Tibor Bisztriczky’s
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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 : .Geometry Fest , Renyi Institute,June 11-15, 2007, Budapest.
.
Canadian
Mathematical Bulletin, 52 (3),September 2009
. Associate Member, Renyi Institue of Mathematics,
2011-14
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© 2002 D.B.