"Critical intensities of Boolean models with different underlying convex shapes",
( with Rahul Roy)
Adv. in Appl. Probab. 34 (2002), no. 1, 48-57
We consider the Poisson Boolean model of percolation where the percolating
shapes are convex regions. By an enhancement argument we strengthen a result
of Jonasson [2000] to show that the critical intensity of percolation in
2-dimensions is minimised among the class of convex shapes of unit area when
the percolating shapes are triangles, and, for any other shape, the critical
intensity is strictly larger than this minimum value. We also obtain a partial
generalisation to higher dimension. In particular, for 3 dimensions,
the critical intensity of percolation is minimised among the class of regular
polytopes of unit volume when the percolating shapes are tetrahedrons.
Moreover, for any other regular polytope the critical intensity is strictly larger
than this minimum value.
