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Balázs Csikós - Some results around the Hadwiger-Kneser-Poulsen conjecture
| BALÁZS CSIKÓS, Eötvös University, Budapest, Rákóczi út 5, H-1088 Hungary |
| Some results around the Hadwiger-Kneser-Poulsen conjecture |
In 1954-56 Poulsen, Kneser and Hadwiger formulated the following
conjecture: Let
and
be
congruent balls in the Euclidean n-space with centers
and
respectively. If
for all
, then the volume of
is not
less
than that of
. Although the conjecture is
still
open, even in the planar case, it can be proved on the additional
condition that one can move the points pi to the points p'icontinuously in such a way that the distances between the points
decrease during the motion. We shall discuss generalizations of this
theorem for balls in the spherical and hyperbolic space and also for
domains obtained from balls by means of the operations 
and
. The proofs are based on some formulae for the variation of the
volume and a suitable modification of the Dirichlet-Voronoi
decomposition. Some applications will also be presented.
Next: Ludwig Danzer and Gerrit Up: Discrete Geometry / Géométrie Previous: H. S. MacDonald Coxeter
