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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
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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

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