On Convex Sets of Multivariate Distributions and Their Extreme Points

Suppose F ₁ and F ₂ are two n-variate distribution functions. It is clear that any convex combination of F ₁ and F ₂ is again an n-variate distribution function. More precisely, if 0≤λ≤1, then the function λF ₁ + (1−λ)F ₂ is also a distribution function. A collection J of n-variate distributions is said to be convex if for any given distributions F ₁ and F ₂ in J, every convex combination of F ₁ and F ₂ is also a member of J. An element F in a convex set J is said to be an extreme point of J if there is no way we can find two distinct distributions F ₁ and F ₂ in J and a number 0<λ<1 such that F=λF ₁ + (1−λ)F ₂.