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 ndimensional Euclidean space R^{n} to random compact sets.
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 nonempty, 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 nonempty, compact subset of R^{n}, let
∥
K
∥
=
max
{
∥
v
∥
R
n

v
∈
K
}
and define the expectation E[X] of X to be
E
[
X
]
=
{
E
[
V
]

V
is a selection of
X
and
E
∥
V
∥
<
+
∞
}
.
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] < +∞,
(
v
o
l
(
E
[
X
]
)
)
1
/
n
≥
E
[
v
o
l
(
X
)
1
/
n
]
,
where "vol" denotes ndimensional Lebesgue measure.
If X takes the values (nonempty, 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.