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Borsuk’s Problem

  • Chuanming Zong
  • James J. Dudziak
Part of the Universitext book series (UTX)

Abstract

Let X denote a subset of R n . 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 R n can be partitioned into n +1 subsets X1, X2,...., Xn+1 such that
$$d\left( {{X_i}} \right) < d\left( X \right), i = 1,2, \ldots ,n + 1?$$

Keywords

Convex Body Affirmative Answer Element Subset Symmetric Convex Body Regular Simplex 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer-Verlag New York, Inc. 1996

Authors and Affiliations

  • Chuanming Zong
    • 1
  • James J. Dudziak
    • 2
  1. 1.Institute of MathematicsThe Chinese Academy of SciencesBeijingPR China
  2. 2.Department of MathematicsBucknell UniversityLewisburgUSA

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