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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 $B_1,\dots, B_N$ and $B'_1,\dots,B'_N$ be congruent balls in the Euclidean n-space with centers $p_1,\dots,p_N$and $p'_1,\dots,p'_N$ respectively. If $p_ip_j\geq p'_ip'_j$ for all $1\leq i<j\leq N$, then the volume of $\bigcup_{i=1}^N B_i$ is not less than that of $\bigcup_{i=1}^N B'_i$. 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 $\cup$ and $\cap$. 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.


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Next: Ludwig Danzer and Gerrit Up: Discrete Geometry / Géométrie Previous: H. S. MacDonald Coxeter
 

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