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

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

and define the set-valued expectation E[X] of X to be

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

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 $E[\|X\|]<+\infty$,

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

where "$vol_{n}$" 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.

References

• 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 (2): 286–293. doi:10.1016/0047-259X(90)90052-J.