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.

[edit] 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

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

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] < +∞,

where "vol" denotes n-dimensional Lebesgue measure.

[edit] 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.

[edit] References

• Gardner, Richard J. (2002). "The Brunn-Minkowski inequality". Bull. Amer. Math. Soc. (N.S.) 39 (3): pp. 355–405 (electronic). doi:10.1090/S0273-0979-02-00941-2. 
• Vitale, Richard A. (1990). "The Brunn-Minkowski inequality for random sets". J. Multivariate Anal. 33: 286–293. doi:10.1016/0047-259X(90)90052-J.