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Convex Sets, Cantor Sets and a Midpoint Property

Published online by Cambridge University Press:  20 November 2018

Harold Reiter
Harold Reiter*
Affiliation:
Dept. of Math., University of North CarolinaUNCC Station, Charlotte, N.C. 28223, U.S.A.
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It is well known that every point of the closed unit interval I can be expressed as the midpoint of two points of the Cantor ternary set D. See [2, p. 549] and [3, p. 105]. Regarding J as a one dimensional compact convex set, it seems natural to try to generalize the above result to higher dimensional convex sets. We prove in section 3 that every convex polytope K in Euclidean space Rd contains a topological copy C of D such that each point of K is expressible as a midpoint of two points of C. Also, we give necessary and sufficient conditions on a planar compact convex set for it to contain a copy of D with the midpoint property above. In the final section we prove a result on minimal midpoint sets.

Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 19 , Issue 4 , 01 December 1976 , pp. 467 - 471
Copyright
Copyright © Canadian Mathematical Society 1976

References

1. Grünbaum, B., Convex Polytopes, Wiley-Interscience (London-New York-Sidney), 1967.Google Scholar
2. Randolph, J. F., Distances between points of the Cantor set, American Mathematical Monthly, 47 (1940).Google Scholar
3. Steinhaus, H., Now a vlasnośc mnogości, G. Cantora, Wektor, 1917.Google Scholar