Today (28 Sep 2015) we made some progress on this question/conjecture:
Project 2. Does the only E_f-critical k-configuration on the circle
S^1 in R^2 consist of k equally-spaced points? [If so, that would give
a result much stronger than the Cohn-Kumar theorem about E_f-minimizers.
As far as I know, this may be an open research problem - you can help us
search the library/web to see what's known!]
[Note added 14 December 2015: Kevin P points out that unless f has more
conditions (beyond completely mononotonic) imposed, there are other
critical configurations which can arise (there may be configurations
where various subsets of points coincide - these appear in Harvey Cohn's
1960 Amer. Math. Monthly paper) though the sense in which these are
"critical" needs further discussion: e.g. for a smooth f(t), being
decreasing and convex with f'(0)=0 forces f to be constant; on the other
hand, remember that t is distance-squared, so criticality of E_f only
requires df/d√t=0 at t=0, and that includes any f with f'(0) bounded.
Of course, I was (tacitly) assuming f(t) and associated energy E_f were
*proper* functions on (0,2] and the configuration space, respectively,
meaning that they tend to ∞ when t tends to 0, i.e where points collide.]
* * *
We considered the special case E=E_f where f(t)=1-t/2=s, i.e. where
g(s)=s. [This is "borderline" because this g (or f) barely satisfies
the usual criteria: although g is not positive, it is bounded below by
-1, so that OK - indeed, we can add a constant to make it positive
without affecting its gradient; g'=1>0, so g is increasing; and g"=0,
so g is convex (again, just barely)]. We did this case because the
weighting factor g'(s) on the gradient terms simplifies to 1, whereas
it usually depends on the particular value of dot (scalar) product s.
* * *
Show (for this special E) that points in a E-critical configuration
must balance: u_1+u_2+u_3+...+u_k = 0. The trick (as we did in class
today for the case k=4) is to express the tangential gradient ∂E=DE^T
using the cross product ∂E=U¢DE, note that D_iE=u_1+...+^u_i^+...+u_k
(^u_i^ means "with u_i omitted"), and thus ∂_iE=u_i¢(u_1+...+u_k)
[since I can't make an ASCII cross product, I've used ¢, i.e. alt-$].
Then criticality implies 0=∂_iE=u_i¢D_iE= u_i¢(u_1+...+u_k) for all i;
and if this holds for two linearly independent vectors (say, WLOG, for
u_1 and u_2), we conclude u_1+...+u_k=0. For k=4, we argued (W 9/30)
geometrically that an balanced U=(u,v,-u,-v), and conversely all of
these are critical for the case g(s)=s, and so uniquess fails (likely
from its "borderline" convexity.
* * *
What happens on S^1 for general g, whose weighting g'(s) varies with
s? [Please try the cases k=3 and k=4 first.] We mechanically
interpreted the criticality condition 0=∂_iE at u_i to mean that we
can "balance" the circle, equipped with weights g'(s_ij) at u_j, along
the "diameter" between u_i and -u_i. We then used this to deduce some
qualitative features of critical configurations in general, e.g. the k
points cannot all lie in (open) hemisphere (i.e. a semicircle).
* * *
We also began to explore whether viewing the energy \E_g as a sum of
the (strictly) convex functions g(s_ij), would itself be a (strictly)
convex function of the angle-cosines or (dot products) s_ij - this
argument could only work for S^1 which is 1-dimensional (and so the
configuration U=(...,u_i,...) is essentially determined by the angle
cosines s_ij). Of course this can't work in higher-dimensional
spheres since that would mean only one critical configuration there,
which is absurd, but if it does work then we would know that the
equally-spaced configuration is the unique critical point (up to
rotation) on S^1.