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 to random compact sets. Statement of the inequality. Let "X" be a random compact set in R; that is, a Borel–measurable function from some probability space (Ω, Σ, Pr) to the space of non-empty, compact subsets of R equipped with the Hausdorff metric. A random vector "V" : Ω → R is called a selection of "X" if Pr("V" ∈ "X") = 1 . If "K" is a non-empty, compact subset of R, let and define the set-valued expectation E["X"] of "X" to be Note that E["X"] is a subset of R. In this notation, Vitale's random Brunn–Minkowski inequality is that, for any random compact set "X" with formula_3, where "formula_5" 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. Vitale's