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1 [made Community Wiki]

I was trying to write this as a comment, but the place is too small.

According to the intro of http://www.jstor.org/stable/1428226, the perimeter of the typical cell is in $\mathbb{R}^2$: $4/\sqrt{\lambda}$ ($\lambda$: intensity of the Poisson point process). It means that if you take a large ball, compute the sum of perimeters of all cells, and divide by the number of cells, you converge to this value. So using ergodicity the average perimeter should be the average number of cells (i.e. of points) multiplied by the perimeter of the typical cell, and divided by $2$ (because each cell is counted twice), which makes $2\sqrt{\lambda}$ (asymptotically). All this is quite vague but in the same intro there are vey good references, which also allow for arbitrary dimension.
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