Geodesic
Continuous Families of Curves
Canadian journal of mathematics, Tome 18 (1966) no. 1, pp. 529-537

Voir la notice de l'article provenant de la source Cambridge University Press

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. 
Grünbaum, Branko. Continuous Families of Curves. Canadian journal of mathematics, Tome 18 (1966) no. 1, pp. 529-537. doi: 10.4153/CJM-1966-052-4
@article{10_4153_CJM_1966_052_4,
     author = {Gr\"unbaum, Branko},
     title = {Continuous {Families} of {Curves}},
     journal = {Canadian journal of mathematics},
     pages = {529--537},
     year = {1966},
     volume = {18},
     number = {1},
     doi = {10.4153/CJM-1966-052-4},
     url = {http://geodesic.mathdoc.fr/articles/10.4153/CJM-1966-052-4/}
}
TY  - JOUR
AU  - Grünbaum, Branko
TI  - Continuous Families of Curves
JO  - Canadian journal of mathematics
PY  - 1966
SP  - 529
EP  - 537
VL  - 18
IS  - 1
UR  - http://geodesic.mathdoc.fr/articles/10.4153/CJM-1966-052-4/
DO  - 10.4153/CJM-1966-052-4
ID  - 10_4153_CJM_1966_052_4
ER  - 
%0 Journal Article
%A Grünbaum, Branko
%T Continuous Families of Curves
%J Canadian journal of mathematics
%D 1966
%P 529-537
%V 18
%N 1
%U http://geodesic.mathdoc.fr/articles/10.4153/CJM-1966-052-4/
%R 10.4153/CJM-1966-052-4
%F 10_4153_CJM_1966_052_4

[1] 1. Bruckner, A. M. and Bruckner, J. B., Generalized convex kernels, Israel J. Math., 2 (1964), 27–32. Google Scholar

[2] 2. Ceder, J., On outwardly simple line families, Can. J. Math., 16 (1964), 1–11. Google Scholar

[3] 3. Ceder, J., On a problem of Grilnbaum, Proc. Amer. Math. Soc, 16 (1965), 188–189. Google Scholar

[4] 4. Forrester, A., A theorem on involutory transformations without fixed points, Proc. Amer. Math. Soc, 3 (1952), 333–334. Google Scholar

[5] 5. Goldberg, M., On area-bisectors of plane convex sets, Amer. Math. Monthly, 70 (1963), 529–531. Google Scholar

[6] 6. Grünbaum, B., Measures of symmetry for convex sets, Proc. Symp. Pure Math., 7 (Convexity) (1963), 233–270. Google Scholar

[7] 7. Hammer, P. C. and Sobczyk, A., Planar line families, I, II, Proc. Amer. Math. Soc, 4 (1953), 226–233, 341-349. Google Scholar

[8] 8. Horn, A. and Valentine, F. A., Seme properties of L-sets in the plane, Duke Math. J., 16 (1949), 131–140. Google Scholar

[9] 9. Menon, V. V., A theorem on partitions of mass-distributions, Pacific J. Math, (to appear). Google Scholar

[10] 10. Piegat, E., O srednicach figur wypuklych plaskich, Roczn. Polsk. Towarz. Mat., Ser. 2, 7 (1963), 51–56. Google Scholar

[11] 11. Smith, T. J., Planar line families, Ph.D. Thesis, University of Wisconsin, 1961. Google Scholar

[12] 12. Steinhaus, H., Quelques applications des principes topologiques à la géométrie des corps convexes, Fund. Math., 41 (1955), 284–290. Google Scholar

[13] 13. Viet, U., Umkehrung eines Satzes von H. Brunn über Mittelpunktseibereiche, Math.-Phys. Semesterber., 5 (1956), 141–142. Google Scholar

[14] 14. Zarankiewicz, K., Bisection of plane convex sets by lines, Wiadom. Mat., Ser. 2, 2 (1959), 228–234 (Polish). Google Scholar

[15] 15. Zindler, K., Über konvexe Gebilde, I, II, III, Monatsh. Math., 30 (1920), 87–102; 31 (1921), 25-56; 32 (1922), 107-138. Google Scholar

Cité par Sources :