Erschienen in:

1996 | OriginalPaper | Buchkapitel

Borsuk’s Problem

verfasst von : Chuanming Zong, James J. Dudziak

Erschienen in: Strange Phenomena in Convex and Discrete Geometry

Verlag: Springer New York

Enthalten in: Professional Book Archive

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Let X denote a subset of Rn. As usual, we call $$d\left( X \right) = \mathop {\sup }\limits_{x,y \in X} \left\| {x - y} \right\|$$ the diameter of X. In studying the relation between a set and its subsets of smaller diameter, K. Borsuk [2] in 1933 raised the following famous problem:Borsuk’s Problem. Is it true that every bounded set X in Rncan be partitioned into n +1 subsetsX₁, X₂,...., X_n+1such that$$d\left( {{X_i}} \right) < d\left( X \right), i = 1,2, \ldots ,n + 1?$$

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