A Poisson allocation of optimal tail

@article{Marko2011APA,
  title={A Poisson allocation of optimal tail},
  author={Roland Mark'o and 'Ad'am Tim'ar},
  journal={arXiv: Probability},
  year={2011}
}
The allocation problem for a $d$-dimensional Poisson point process is to find a way to partition the space to parts of equal size, and to assign the parts to the configuration points in a measurable, "deterministic" (equivariant) way. The goal is to make the diameter $R$ of the part assigned to a configuration point have fast decay. We present an algorithm for $d\geq3$ that achieves an $O(\operatorname {exp}(-cR^d))$ tail, which is optimal up to $c$. This improves the best previously known… 
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