Cosine sum problem Consider any n points on the circumference of a circle. Draw a straight line link between each pair of points with a link weight consisting of the cosine of the angle the link subtends at the centre.
It seems that If the convex hull of the point set contains the centre of the circle, then some point has the property that the sum of the link weights that meet that point are less than or equal to -1.
I have verified for the cases n=3 or 4 nodes and for many simulations - but cannot prove in general. Would be glad of any help
 A: Center the circle at the origin.  Consider the sum of all the points.  If it is (0,0) then the statement is true for all points.  Otherwise there is an arc of pi in which the sum has an x value less than 0.  Since the condition of the convex hull enclosing (0,0) is equivalent to no two adjacent points being separated by more than pi, we are done (by the pigeonhole principle).
A: Assume $\sum_{i=1}^n a_i x_i=0$ with $x_i$ on the circle and $a_i>0$, $\sum a_i=1$.
Then
$\sum_{i=1}^n\sum_{j=1}^n a_i\langle x_i,x_j\rangle=0$.
It follows that for some $i$, $\sum_{j=1}^n\langle x_i,x_j\rangle\le 0$.
Since $\langle x_i,x_i\rangle=1$, it implies
$\sum_{j; j\not=i} \langle x_i,x_j\rangle\le -1$ which is what you want.
A: This solution isn't much different at bottom from those of GS and DB but it's how I think of it. Let the circle be the unit circle in the complex plane. Let the points be $z_1,\dots,z_n$. The "link weight" of the chord joining $z_j$ to $z_k$ is the real part of $\overline{z}_jz_k$ (I omit the small amount of calculation needed to verify this). Let $S$ be the sum of all the $z_j$. Then the sum of the link weights at $z_j$ is the sum on all $k$ other than $j$ of the real part of $\overline{z}_jz_k$, which is the real part of ($\overline{z}_j$ times the sum of the $z_k$), which is ${\rm Re}(\overline{z}_j(S-z_j))={\rm Re}(\overline{z}_jS)-1$. If $S=0$, we're done; the sum of the link weights at $z_j$ is $-1$. If $S\ne0$, then the $z$ such that the real part of $\overline{z}S$ is positive form an open half plane with the origin on the boundary. The points can't all be in that open half plane, since the origin is in their convex hull. So at least one of the points $z_j$ has real part of $\overline{z}_jS$ at most zero, and we win. 
A: There seems to be some misunderstanding re the problem, which I posted. The differences in the x coordinates between points are not equal or proportional to the link weights since the chord lengths between points vary.
This seems to be a very tricky problem probably requiring variational methods or the like.
