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Arcs with increasing chords

Published online by Cambridge University Press:  24 October 2008

D. G. Larman  and
P. McMullen
D. G. Larman
Affiliation:
University College London, Gower Street, London WC1E 6BT
P. McMullen
Affiliation:
University College London, Gower Street, London WC1E 6BT
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Let f:[0, 1]→R2 be a Jordan arc, and for t, u ∈ [0, 1] let d(t, u) = d(f(t), f(u)) denote the Euclidean length of the chord between f(t) and f(u), and l(t, u) = l(f(t), f(u)) the corresponding arc-length, when this is defined. We say that f has the increasing chord property if d(t2, t3) ≤ d(t1, t4) whenever 0 ≤ t1 ≤ t2 ≤ t3 ≤ t4 ≤ 1. In connexion with a problem in complex analysis, K. Binmore has asked (private communication, see (1)) whether there exists an absolute constant K such that

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Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 72 , Issue 2 , September 1972 , pp. 205 - 207
Copyright
Copyright © Cambridge Philosophical Society 1972

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References

REFERENCES

(1)Binmore, K.On Turan's lemma. Bull. London Math. Soc. 3 (1971), 313–317.Google Scholar