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Continuous Families of Curves

Published online by Cambridge University Press:  20 November 2018

Branko Grünbaum*
Affiliation:
Hebrew University, Jerusalem and Michigan State University
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The present paper is an attempt to find the unifying principle of results obtained by different authors and dealing—in the original papers—with areabisectors, chords, or diameters of planar convex sets, with outwardly simple planar line families, and with chords determined by a fixed-point free involution on a circle. The proofs in the general setting seem to be simpler and are certainly more perspicuous than many of the original ones. The tools required do not transcend simple continuity arguments and the Jordan curve theorem. The author is indebted to the referee for several helpful remarks.

Type
Research Article
Copyright
Copyright © Canadian Mathematical Society 1966

References

1. Bruckner, A. M. and Bruckner, J. B., Generalized convex kernels, Israel J. Math., 2 (1964), 2732.Google Scholar
2. Ceder, J., On outwardly simple line families, Can. J. Math., 16 (1964), 111.Google Scholar
3. Ceder, J., On a problem of Grilnbaum, Proc. Amer. Math. Soc, 16 (1965), 188189.Google Scholar
4. Forrester, A., A theorem on involutory transformations without fixed points, Proc. Amer. Math. Soc, 3 (1952), 333334.Google Scholar
5. Goldberg, M., On area-bisectors of plane convex sets, Amer. Math. Monthly, 70 (1963), 529531.Google Scholar
6. Grünbaum, B., Measures of symmetry for convex sets, Proc. Symp. Pure Math., 7 (Convexity) (1963), 233270.Google Scholar
7. Hammer, P. C. and Sobczyk, A., Planar line families, I, II, Proc. Amer. Math. Soc, 4 (1953), 226233, 341-349.Google Scholar
8. Horn, A. and Valentine, F. A., Seme properties of L-sets in the plane, Duke Math. J., 16 (1949), 131140.Google Scholar
9. Menon, V. V., A theorem on partitions of mass-distributions, Pacific J. Math, (to appear).Google Scholar
10. Piegat, E., O srednicach figur wypuklych plaskich, Roczn. Polsk. Towarz. Mat., Ser. 2, 7 (1963), 5156.Google Scholar
11. Smith, T. J., Planar line families, Ph.D. Thesis, University of Wisconsin, 1961.Google Scholar
12. Steinhaus, H., Quelques applications des principes topologiques à la géométrie des corps convexes, Fund. Math., 41 (1955), 284290.Google Scholar
13. Viet, U., Umkehrung eines Satzes von H. Brunn über Mittelpunktseibereiche, Math.-Phys. Semesterber., 5 (1956), 141142.Google Scholar
14. Zarankiewicz, K., Bisection of plane convex sets by lines, Wiadom. Mat., Ser. 2, 2 (1959), 228234 (Polish).Google Scholar
15. Zindler, K., Über konvexe Gebilde, I, II, III, Monatsh. Math., 30 (1920), 87102; 31 (1921), 25-56; 32 (1922), 107-138.Google Scholar