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 (&Omega;, &Sigma;, Pr) to the space of non-empty, compact subsets of Rn equipped with the Hausdorff metric. A random vector V : &Omega; &rarr; Rn is called a selection of X if Pr(V &isin; 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 &minus; &lambda; and &lambda; respectively, then Vitale's random Brunn–Minkowski inequality is simply the original Brunn–Minkowski inequality for compact sets.