# Absolutely convex set

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

## 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 $\lambda_1 x_1 + \lambda_2 x_2$ 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 light gray area is the Absolutely convex hull of the cross.

The absolutely convex hull of the set A assumes the following representation

$\mbox{absconv} A = \left\{\sum_{i=1}^n\lambda_i x_i : n \in \N, \, x_i \in A, \, \sum_{i=1}^n|\lambda_i| \leq 1 \right\}$.

## References

• Robertson, A.P.; W.J. Robertson (1964). Topological vector spaces. Cambridge Tracts in Mathematics 53. Cambridge University Press. pp. 4–6.