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Günter Rote:

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Curves with increasing chords

*Mathematical Proceedings of the Cambridge Philosophical Society 115
(1994), 1-12, (Zbl 802.51023).*
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Abstract

A curve has increasing chords if *AD* ≥ *BC* for any four points*A*,*B*,*C*,*D*
lying on the curve in that order. The length of such a curve that connects
two points at distance 1 is at most 2*pi/3 in two dimensions, which is
the optimal bound, and less that 30 in three dimensions.

A related problem concerns the length of generalized
self-approaching curves.
Last update: March 26, 2001.