Abstract
Letbe a Poisson process of intensity one in the infinite plane ℝ2. We surround each pointxofby the open disc of radiusrcentred atx. Now letSnbe a fixed disc of arean, and letCr(Sn) be the set of discs which intersectSn. WriteErkfor the event thatCr(Sn) is ak-cover ofSn, andFrkfor the event thatCr(Sn) may be partitioned intokdisjoint single covers ofSn. We prove that P(Erk∖Frk) ≤ck/ logn, and that this result is best possible. We also give improved estimates for P(Erk). Finally, we study the obstructions tok-partitionability in more detail. As part of this study, we prove a classification theorem for (deterministic) covers of ℝ2with half-planes that cannot be partitioned into two single covers.
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