# co.combinatorics – Happy ants never leave compact domain?

I am curious if the following seemingly simple question has an easy answer?

Consider an ant population of $$N$$ ants that lives in $$mathbb R^2$$. Each ant can be considered to be a disk or radius $$1.$$ Ants like to be close to their peers but also not too close. The optimal distance between the center of two ants is $$5^{2/3}$$. So given two coordinates of centers of ants $$x_i,x_j in mathbb R^2$$. Their happiness $$H$$ is $$H(x_i,x_j):=-vert vert x_i-x_j vert^{3/2} -5 vert.$$ Distances $$vert x_i-x_jvert le 1$$ are not allowed since ants cannot be on top of each other.

Now consider the total happiness $$H_N:=sum_{i The question is the following:

Consider any maximiser $$x=(x_1,..,x_N)$$ of happiness $$H_N.$$ Can we show that any optimizer is always contained in a ball $$B(y(x),rsqrt{N})$$ where $$r$$ is a universal positive constant independent of $$N$$ and the maximiser and $$y$$ is allowed to depend on $$x$$. That means, all $$x_i$$ of the optimizer are in $$B(y(x),rsqrt{N}).$$

The conjecture of $$B(y(x),rsqrt{N})$$ is due to the fact that order $$N$$ lattice particles would fit into a ball of radius $$B(y(x),rsqrt{N}).$$

The question sounds almost like a school problem, but I just cannot exclude weird configurations.

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