# 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 Borelmeasurable 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

${\displaystyle \|K\|=\max \left\{\left.\|v\|_{\mathbb {R} ^{n}}\right|v\in K\right\}}$

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

${\displaystyle \mathrm {E} [X]=\{\mathrm {E} [V]|V{\mbox{ is a selection of }}X{\mbox{ and }}\mathrm {E} \|V\|<+\infty \}.}$

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] < +∞,

${\displaystyle \left(\mathrm {vol} \left(\mathrm {E} [X]\right)\right)^{1/n}\geq \mathrm {E} \left[\mathrm {vol} (X)^{1/n}\right],}$

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.