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Let $C$ be a circle of radius $1$ in the complex plane with $n$ points on the boundary. Provide an upper bound on the product of the distances of a given point on the circle to the other $n$ points. The goal is to provide an answer based on how densely the point are positioned on the circle.

The goal is to characterize this based on how densely the points are placed. I am not sure of the correct notion of density to use for the unit circle but on the real line the correct notion would be something like this first described by Arne Beurling: For a discrete set on the real line Λ denote by n(r) the smallest number of points in any interval [x,x+r], r>0. The lower uniform density is defined by $l.u.d.(\Lambda)=\lim_{r\rightarrow\infty}\frac{n(r)}{r}$.

So the goal is to give an answer of the form

maximum product of distances $\le$ a function of the correct notion of l.u.d. on the circle.

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I think you should mention something about the bounds that you already know. To avoid getting obvious bounds like $2^n$ –  Amr Jun 12 '13 at 18:38
    
Well, there is $2^n$, which cannot be improved on. So we need another parameter. –  André Nicolas Jun 12 '13 at 18:39
    
The goal is to characterize this based on how densely the points are places. For example imagine that the distance between any two of the n points is bigger than $\Delta$. I am not sure of the correct notion of density to use for the unit circle but on the real line the correct notion would be something like this first described by Arne Beurling: For a discrete set on the real line $\Lambda$ denote by n(r) the smallest number of points in any interval [x,x+r], r>0. The lower uniform density is defined by l.u.d.($\Lambda$)$=\lim_{r to \infinity}\frac{n(r)}{r}$ –  mohi Jun 12 '13 at 18:49
    
you may want to look at mathoverflow.net/questions/63525 and mathoverflow.net/questions/64099/… –  leshik Jun 12 '13 at 18:53

1 Answer 1

Every distance is less than or equal to 2 which means an upper will be $2^n$.Note that it is achieved when all points except one point is diametrically opposite to that one point.

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