Absolutely convex set

In mathematics, a subset "C" of a real or complex vector space is said to be absolutely convex or disked if it is convex and balanced (some people use the term "circled" instead of "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 formula_1 is a subset of a real or complex vector space formula_2 then we call formula_1 a disk and say that formula_1 is disked, absolutely convex, and convex balanced if any of the following equivalent conditions is satisfied: 
Recall that the smallest convex (resp. balanced) subset of formula_5 containing a set is called the convex hull (resp. balanced hull) of that set and is denoted by formula_6 (resp. formula_7). 
Similarly, we define the disked hull, the absolute convex hull, or the convex balanced hull of a set formula_1 is defined to be the smallest disk (with respect to subset inclusion) containing formula_9 
The disked hull of formula_1 will be denoted by formula_11 or formula_12 and it is equal to each of the following sets: 
Properties.
Examples.
Although formula_13 the convex balanced hull of formula_1 is necessarily equal to the balanced hull of the convex hull of formula_9 
For an example where formula_16 let formula_5 be the real vector space formula_18 and let formula_19 
Then formula_20 is a strict subset of formula_12 that is not even convex. 
In particular, this example also shows that the balanced hull of a convex set is necessarily convex. 
To see this, note that formula_12 is equal to the closed square in formula_5 with vertices formula_24 and formula_25 while formula_20 is a closed "hour glass shaped" shaped subset that intersects the formula_27-axis at the origin and is the union of two triangles: one whose vertices are the origin along with formula_1 and the other triangle whose vertices are the origin along with formula_29 