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 $C$ 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

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

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. Vol. 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.

• Schaefer, H.H. (1999). Topological vector spaces. Springer-Verlag Press. p. 39.