Borsuk’s Problem

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


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
$$d\left( {{X_i}} \right) < d\left( X \right), i = 1,2, \ldots ,n + 1?$$


Convex Body Affirmative Answer Element Subset Symmetric Convex Body Regular Simplex 
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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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