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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

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

and define the set-valued 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\|]<+\infty,

:\left( \mathrm{vol}_n \left( \mathrm{E} [X] \right) \right)^{1/n} \geq \mathrm{E} \left[ \mathrm{vol}_n (X)^{1/n} \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

  • {{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=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

| doi-access=free

}}

  • {{cite journal

| last = Vitale

| first = Richard A.

| title = The Brunn-Minkowski inequality for random sets

| journal = J. Multivariate Anal.

| volume = 33

| issue = 2

| year = 1990

| pages = 286–293

| doi = 10.1016/0047-259X(90)90052-J

| doi-access = free

}}

{{Lp spaces}}

{{Measure theory}}

{{DEFAULTSORT:Vitale's random Brunn-Minkowski inequality}}

Category:Probabilistic inequalities

Category:Theorems in measure theory