Re: Radius of largest n+1 balls in n-dimensional unit cube?

Thank you Quasi! Would you please provide a hint about how I can
derive the formula you proposed? The only piece I understand is that
sqrt(n) is maximal distance between two corners of the cube, but have
no idea how it entered into your formula.


On Jan 13, 2:01 pm, quasi <qu...@xxxxxxxx> wrote:
On Tue, 13 Jan 2009 11:19:11 -0800 (PST), Golabi Doon

<golabid...@xxxxxxxxx> wrote:
Hello,

I would appreciate your help or comment about the following problem.
Consider a N dimensional space. If I want to put N+1 balls (all with
the same radius R), within the unit hypercube such that:

1. The balls do not cut through each other
2. One of the balls is at the center of the cube, i.e. at (0.5 , 0.5,
0.5, ...., 0.5)

Then what is the maximum possible R in terms of N? If not easy, a good
approximation will be helpful too.

It seems intuitive that the maximum r for your problem occurs when the
other n balls are in the corners, tangent to the central ball and
tangent to the corner faces. For that configuration,

   r = sqrt(n) / (2*(2+sqrt(n)))

But note, the unit cube in R^n has 2^n corners, so you could just as
easily have (2^n)+1 balls with the same radius as above, rather than
only n+1.

quasi

.

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