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.
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\left\{\left.\|v
| \| | |
| Rn |
\right|v\inK\right\}
and define the set-valued expectation E[''X''] of X to be
E[X]=\{E[V]|VisaselectionofXandE\|V\|<+infty\}.
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\|]<+infty
\left(voln\left(E[X]\right)\right)1/n\geqE\left[voln(X)1/n\right],
where "
voln
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.