Günter Rote:

Curves with increasing chords

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

Abstract

A curve has increasing chords if ADBC for any four pointsA,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.
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Last update: March 26, 2001.