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
- [math]\displaystyle{ \| K \| = \max \left\{ \left. \| v \|_{\mathbb{R}^{n}} \right| v \in K \right\} }[/math]
and define the set-valued expectation E[X] of X to be
- [math]\displaystyle{ \mathrm{E} [X] = \{ \mathrm{E} [V] | V \mbox{ is a selection of } X \mbox{ and } \mathrm{E} \| V \| \lt + \infty \}. }[/math]
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 [math]\displaystyle{ E[\|X\|]\lt +\infty }[/math],
- [math]\displaystyle{ \left( \mathrm{vol}_n \left( \mathrm{E} [X] \right) \right)^{1/n} \geq \mathrm{E} \left[ \mathrm{vol}_n (X)^{1/n} \right], }[/math]
where "[math]\displaystyle{ vol_n }[/math]" 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". Bull. Amer. Math. Soc. (N.S.) 39 (3): 355–405 (electronic). doi:10.1090/S0273-0979-02-00941-2. http://www.ams.org/bull/2002-39-03/S0273-0979-02-00941-2/S0273-0979-02-00941-2.pdf.
- 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.
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