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