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I consider points in the two-dimensional plane.

An equilateral triangle is a set of three points in the plane which are equidistant.

Suppose now I have $n$ points $x_1,...,x_n$. What is the configuration which minimizes: $$H(x):=\sum_{i,j} (a-\|x_i-x_j\|^2)^2$$ where $a$ is positive real number.

Clearly, if $n=3$, one recovers the equilateral triangle. Could you draw the solution for larger $n$ ? What is the value of $H_{min}$ as a function of $n$ ?

Thanks !

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This is the problem of determing where to place the explosives around the core of an atom bomb, to most efficiently trigger the reaction. It's a classical problem with a substantial literature. Have you looked into it? Generally the minima is not known but in dimension three there's certainly a lot known. In higher dimensions there's quite a bit known for low numbers of points, but again it gets difficult to large numbers of points. –  Ryan Budney Mar 5 '13 at 19:08
@Ryan Budney : Thanks a lot for this idea. Could you give me a reference ? –  user16215 Mar 6 '13 at 7:38
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