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.

Statement 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\$

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

:\mathrm = \.

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\, +\infty,

:\left( \mathrm_n \left( \mathrm \right) \right)^ \geq \mathrm \left \mathrm_n (X)^ \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

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Probabilistic inequalities
Theorems in measure theory