Vitale's random Brunn–Minkowski inequality
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.
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.
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.
- Gardner, Richard J. (2002). "The Brunn-Minkowski inequality" (PDF). Bull. Amer. Math. Soc. (N.S.). 39 (3): 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.