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. The disked hull or the absolute convex hull of a set is the intersection of all disks containing that set.

Properties

If C is a subset of a real or complex vector space X, then the following are equivalent:

1. C is a disk.
2. for any points $x_{1},\,x_{2}$ in C and any scalars a and b satisfying |a| + |b| ≤ 1, the sum $ax_{1}+bx_{2}$ belongs to C.
3. for all scalars a, b, and c satisfying |a| + |b| ≤ |c|, aC + bC ⊆ cC.

The intersection of arbitrarily many absolutely convex sets is again absolutely convex; however, unions of absolutely convex sets need not be absolutely convex anymore.

The disked hull of a set is equal to the convex hull of the balanced hull of that set.

Absolutely convex hull

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

Since the intersection of any collection of absolutely convex sets is absolutely convex, one can define for any subset A of a vector space its absolutely convex hull as the intersection of all absolutely convex sets containing A, analogous to the well-known construction of the convex hull.

More explicitly, one can define the absolutely convex hull of the set A via

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

where the λ_i are elements of the underlying field.

The absolutely convex hull of a bounded set in a topological vector space is again bounded.

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.