Gravitational allocation to Poisson points | Annals of Mathematics

Abstract For d≥3, we construct a non-randomized, fair, and translation-equivariant allocation of Lebesgue measure to the points of a standard Poisson point process in ℝd, defined by allocating to each of the Poisson points its basin of attraction with respect to the flow induced by a gravitational force field exerted by the points of the Poisson process. We prove that this allocation rule is economical in the sense that the allocation diameter, defined as the diameter X of the basin of attraction containing the origin, is a random variable with a rapidly decaying tail. Specifically, we have the tail bound ℙ(X>R)≤Cexp[−cR(logR)αd] for all R>2, where: αd=d−2d for d≥4; α3 can be taken as any number less than −4/3; and C and c are positive constants that depend on d and αd. This is the first construction of an allocation rule of Lebesgue measure to a Poisson point process with subpolynomial decay of the tail ℙ(X>R).

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数学年刊(Annals of Mathematics)

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