Vitale's random Brunn-Minkowski inequality

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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.

[edit] 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 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] < +∞,

\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.

[edit] 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.

[edit] References