Absolutely convex set

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A set C in a real or complex vector space is said to be absolutely convex if it is convex and balanced.

[edit] Properties

A set $C$ is absolutely convex if and only if for any points $x_1, \, x_2$ in $C$ and any numbers $\lambda_1, \, \lambda_2$ satisfying $|\lambda_1| + |\lambda_2| \leq 1$ 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.