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 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 = 1. If K is a non-empty, compact subset of Rn, let
and define the set-valued expectation E of X to be
Note that E is a subset of Rn. In this notation, Vitale's random Brunn–Minkowski inequality is that, for any random compact set X with,
where "" denotes n-dimensional Lebesgue measure.

Relationship to the Brunn–Minkowski inequality

If X takes the values 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.