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Vitale's random Brunn–Minkowski inequality

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In mathematics, Vitale's random Brunn–Minkowski inequality is a theorem due to Richard Vitale that generalizes the classical Brunn–Minkowski inequality for compact subsets of n-dimensional Euclidean space Rn to random compact sets.

Contents

Statement of the inequality

Let X be a random compact set in Rn; that is, a Borel–measurable function from some probability space (Ω, Σ, Pr) to the space of non-empty, compact subsets of Rn equipped with the Hausdorff metric. A random vector V : Ω → Rn is called a selection of X if Pr(V ∈ X) = 1. If K is a non-empty, compact subset of Rn, let

K = max { v R n | v K }

and define the expectation E[X] of X to be

E [ X ] = { E [ V ] | V  is a selection of  X  and  E V < + } .

Note that E[X] is a subset of Rn. In this notation, Vitale's random Brunn–Minkowski inequality is that, for any random compact set X with E[X] < +∞,

( v o l ( E [ X ] ) ) 1 / n E [ v o l ( X ) 1 / n ] ,

where "vol" denotes n-dimensional Lebesgue measure.

Relationship to the Brunn–Minkowski inequality

If X takes the values (non-empty, compact sets) K and L with probabilities 1 − λ and λ respectively, then Vitale's random Brunn–Minkowski inequality is simply the original Brunn–Minkowski inequality for compact sets.

References

Vitale's random Brunn–Minkowski inequality Wikipedia


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