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, in which case it is called a disk.

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

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

See also

• vector (geometric), for vectors in physics
• Vector field

References

• Robertson, A.P.; W.J. Robertson (1964). Topological vector spaces. Cambridge Tracts in Mathematics 53. Cambridge University Press. pp. 4–6.
• Narici, Lawrence; Beckenstein, Edward (July 26, 2010). Topological Vector Spaces, Second Edition. Pure and Applied Mathematics (Second ed.). Chapman and Hall/CRC.