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

Definition

If is a subset of a real or complex vector space, then we call a disk and say that is disked, absolutely convex, and convex balanced if any of the following equivalent conditions is satisfied:

  1. is convex and balanced;
  2. for any scalars and satisfying, ;
  3. for all scalars,, and satisfying |a| + |b| ≤ |c|, ;
  4. for any scalars satisfying, ;
  5. for any scalars, satisfying, ;
Recall that the smallest convex subset of containing a set is called the convex hull of that set and is denoted by .
Similarly, we define the disked hull, the absolute convex hull, or the convex balanced hull of a set is defined to be the smallest disk containing.
The disked hull of will be denoted by or and it is equal to each of the following sets:

  1. , which is the convex hull of the balanced hull of ; thus, ;
    • Note however that in general,, even in finite dimensions.
  2. the intersection of all disks containing

  3. where the are elements of the underlying field.

Sufficient conditions

Properties

Examples

Although, the convex balanced hull of is not necessarily equal to the balanced hull of the convex hull of.
For an example where, let be the real vector space and let.
Then is a strict subset of cobal that is not even convex.
In particular, this example also shows that the balanced hull of a convex set is not necessarily convex.
To see this, note that is equal to the closed square in with vertices, and while is a closed "hour glass shaped" shaped subset that intersects the -axis at the origin and is the union of two triangles: one whose vertices are the origin along with and the other triangle whose vertices are the origin along with.