# Absolutely convex set

A set C in a real or complex vector space is said to be absolutely convex or disked if it is convex and balanced (circled), in which case it is called a disk.

## Properties

A set ${\displaystyle C}$ is absolutely convex if and only if for any points ${\displaystyle x_{1},\,x_{2}}$ in ${\displaystyle C}$ and any numbers ${\displaystyle \lambda _{1},\,\lambda _{2}}$ satisfying ${\displaystyle |\lambda _{1}|+|\lambda _{2}|\leq 1}$ the sum ${\displaystyle \lambda _{1}x_{1}+\lambda _{2}x_{2}}$ belongs to ${\displaystyle C}$.

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 light gray area is the absolutely convex hull of the cross.

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

${\displaystyle {\mbox{absconv}}A=\left\{\sum _{i=1}^{n}\lambda _{i}x_{i}:n\in \mathbb {N} ,\,x_{i}\in A,\,\sum _{i=1}^{n}|\lambda _{i}|\leq 1\right\}}$.