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 R^"n" to random compact sets.

tatement of the inequality

Let "X" be a random compact set in R^"n"; that is, a Borel-measurable function from some probability space (Ω, Σ, Pr) to the space of non-empty, compact subsets of R^"n" equipped with the Hausdorff metric. A random vector "V" : Ω → R^"n" is called a selection of "X" if Pr("V" ∈ "X") = 1. If "K" is a non-empty, compact subset of R^"n", let

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

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

:

Note that E ["X"] is a subset of R^"n". In this notation, Vitale's random Brunn-Minkowski inequality is that, for any random compact set "X" with E ["X"] < +∞,

:$left(\; mathrm\{vol\}\; left(\; mathrm\{E\}\; [X]\; ight)\; ight)^\{1/n\}\; geq\; mathrm\{E\}\; left\; [\; mathrm\{vol\}\; (X)^\{1/n\}\; ight]\; ,$

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.

References

* cite journal
last=Gardner
first=Richard J.
title=The Brunn-Minkowski inequality
journal=Bull. Amer. Math. Soc. (N.S.)
volume=39
issue=3
year=2002
pages=pp. 355–405 (electronic)
url = http://www.ams.org/bull/2002-39-03/S0273-0979-02-00941-2/S0273-0979-02-00941-2.pdf
doi=10.1090/S0273-0979-02-00941-2
* cite journal
last = Vitale
first = Richard A.
title = The Brunn-Minkowski inequality for random sets
journal = J. Multivariate Anal.
volume = 33
year = 1990
pages = 286–293
doi = 10.1016/0047-259X(90)90052-J