Hostname: page-component-586b7cd67f-2brh9 Total loading time: 0 Render date: 2024-11-20T06:44:09.020Z Has data issue: false hasContentIssue false
  • English
  • Français

Sylvester's Problem for Spreads of Curves

Published online by Cambridge University Press:  20 November 2018

Kym S. Watson
Kym S. Watson*
Affiliation:
Australian National University, Canberra, Australia
Rights & Permissions [Opens in a new window]

Extract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Spreads of curves were introduced by Grunbaum in [1]. A spread of curves is a continuous family of simple arcs in the real plane, every two of which intersect in exactly one point. A spread is the continuous analogue of a finite arrangement of pseudolines in the plane. Sylvester's problem for finite arrangements of pseudolines asks if every non-trivial arrangement has a simple vertex, that is a point contained in exactly two pseudolines of the arrangement. This question was answered in the affirmative by Kelly and Rottenberg [5]. One interesting feature of this result is that it does not depend on the pseudolines being straight lines.

Here we settle Sylvester's problem for spreads. We show that every nontrivial spread of line segments has uncountably many simple vertices. But we also give examples of non-trivial spreads with no simple vertices.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 32 , Issue 1 , 01 February 1980 , pp. 219 - 239
Copyright
Copyright © Canadian Mathematical Society 1980

References

1. Grünbaum, B., Continuous families of curves, Can. J. Math. 18 (1966), 529–537.Google Scholar
2. Grünbaum, B., Arrangements and spreads, Conference Board of the Mathematical Sciences, Regional Conference Series in Mathematics, No. 10, (American Mathematical Society, Providence, R.I., 1972).Google Scholar
3. Hammer, P. C. and Sobczyk, A., Planar line families II, Proc. Amer. Math. Soc. 4 (1953), 341–349.Google Scholar
4. Ivan, C., On spreads of curves, Atti. Accad. naz. Lincei, VIII. Ser., Rend., Cl. Sci. fis. mat. natur. 55 (1973), 46–49 (1974).Google Scholar
5. Kelly, L. M. and Rottenberg, R., Simple points in pseudoline arrangements, Pac. J. Math. 40 (1972), 617–622.Google Scholar
6. Zamfirescu, T., On planar continuous families of curves, Can. J. Math. 21 (1969), 513–530.Google Scholar
7. Zamfirescu, T., Les courbes fermées doubles sans points triples associées a une famille continue, Israel J. Math. 7 (1969), 69–89.Google Scholar
8. Zamfirescu, T., Sur les familles continues de courbes. V., Atti. Accad. naz. Lincei, VIII. Ser., Rend., Cl. Sci. fis. mat. natur. 03 (1972), 505–507 (1973).Google Scholar