Absolutely convex set

Properties

A set C is absolutely convex if and only if for any points x 1 , x 2 in C and any numbers λ 1 , λ 2 satisfying | λ 1 | + | λ 2 | ≤ 1 the sum λ 1 x 1 + λ 2 x 2 belongs to C .

Since the intersection of any collection of absolutely convex sets is absolutely convex then for any subset A of a vector space one can define its absolutely convex hull to be the intersection of all absolutely convex sets containing A.

Absolutely convex hull

The absolutely convex hull of the set A is defined to be

absconv A = { ∑ i = 1 n λ i x i : n ∈ N , x i ∈ A , ∑ i = 1 n | λ i | ≤ 1 } .