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On Planar Continuous Families of Curves

Published online by Cambridge University Press:  20 November 2018

Tudor Zamfirescu
Tudor Zamfirescu*
Affiliation:
Institutul de Matematică, Bucharest, Romania Ruhr-Universität Bochum, Bochum, West Germany
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In a recent paper (3), Gránbaum has found a general and unifying setting for a number of properties of some special lines associated with a planar convex body. Besides various interesting results, two conjectures are stated and two kinds of convexity and polygonal connectedness are introduced.

In the present paper, we shall prove one of Gránbaum's conjectures (§ 3, Theorem 1); we consider the other in § 4 and establish some related results in §§ 5 and 6. Six-partite problems are studied in this general setting (§ 7) and a question raised by Ceder (2) is answered. We give a generalization of the notion of a continuous family of curves in § 8, and discuss some new kinds of connectedness in § 9.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 21 , 1969 , pp. 513 - 530
Copyright
Copyright © Canadian Mathematical Society 1969

References

1. Bruckner, A. M. and Bruckner, J. B., Generalized convex kernels, Israel J. Math. 2 (1964), 27–32.Google Scholar
2. Ceder, J., Generalized sixpartite problems, Bol. Soc. Mat. Mexicana 9 (1964), 28–32.Google Scholar
3. Grunbaum, B., Continuous families of curves, Can. J. Math. 18 (1966), 529–537.Google Scholar
4. Horn, A. and Valentine, F. A., Some properties ofL-sets in the plane, Duke Math. J. 16 (1949), 131–140.Google Scholar
5. Zamfirescu, T., Sur les familles continues de courbes (Note I), Rend. Classe Sci. Fis. Mat. Naturali Accad. Naz. dei Lincei 42 (1967), 771–774.Google Scholar