I'm not sure I understand the part about mandating the optima in advance.
If the tangles are well-tangled in space, I'm not sure how an optimum could
be able to be a circle arc.
(Further, aren't tangles supposed to happen *inside* something, like a cube?)
—Dan
Jim Propp wrote:
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Consider the points A=(1,0,0), B=(-1,0,0), C=(0,1,0), and D=(0,-1,0). There
are lots of ways to draw two disjoint 3D arcs joining A with B and C with
D. What if we want them to be short without getting too close to each
other? We might set up some “short-but-not-too-close” objective functional
and optimize it using calculus of variations. There are probably many ways
to do this, some solvable and some not. Does anyone know of work along
these lines?
In the spirit of calculus-of-variations-of-variations, we might mandate the
optima in advance (each arc is a half-circle on the sphere x^2+y^2+z^2=1,
one in the upper hemisphere and one in the lower hemisphere) and then ask,
For what objective functional is this pair of arcs locally optimal?
...
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