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

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

S is convex and balanced;
for any scalars a and b satisfying |a| + |b| ≤ 1, aS + bS ⊆ S;
for all scalars a, b, and c satisfying |a| + |b| ≤ |c|, aS + bS ⊆ cS;
for any scalars a1, ..., an satisfying \( {\displaystyle \sum _{i=1}^{n}|a_{i}|\leq 1}\) , \( {\displaystyle a_{1}S+\cdots +a_{n}S\subseteq S} \);
for any scalars c, a1, ..., an satisfying \( {\displaystyle \sum _{i=1}^{n}|a_{i}|\leq |c|} \) , \( {\displaystyle a_{1}S+\cdots +a_{n}S\subseteq cS} \);

Recall that the smallest convex (resp. balanced) subset of X containing a set is called the convex hull (resp. balanced hull) of that set and is denoted by co(S) (resp. bal(S)).

Similarly, we define the disked hull, the absolute convex hull, or the convex balanced hull of a set S is defined to be the smallest disk (with respect to subset inclusion) containing S.[1] The disked hull of S will be denoted by disk S or cobal S and it is equal to each of the following sets:

co(bal(S)), which is the convex hull of the balanced hull of S; thus, cobal(S) = co(bal(S));
Note however that in general, cobal(S) ≠ bal(co(S)), even in finite dimensions.
the intersection of all disks containing S
\( {\displaystyle \left\{\sum _{i=1}^{n}\lambda _{i}x_{i}:n\in \mathbb {N} ,\,x_{i}\in A,\,\sum _{i=1}^{n}|\lambda _{i}|\leq 1\right\},} \) where the λi are elements of the underlying field.

Sufficient conditions

The intersection of arbitrarily many absolutely convex sets is again absolutely convex; however, unions of absolutely convex sets need not be absolutely convex anymore.
if D is a disk in X, then X is absorbing in X if and only if span D = X.[2]

Properties
See also: Topological vector space § Properties

If S is an absorbing disk in a vector space X then there exists an absorbing disk E in X such that E + E ⊆ S.^[3]
The convex balanced hull of S contains both the convex hull of S and the balanced hull of S.
The absolutely convex hull of a bounded set in a topological vector space is again bounded.
If D is a bounded disk in a TVS X and if x_• = (x_i)^∞
_i=1 is a sequence in D, then the partial sums s_• = (s_n)^∞
_i=1 are Cauchy, where for all n, s_n := ∑ⁿ
_i=1 2^-i x_i .^[4]
- In particular, if in addition D is a sequentially complete subset of X, then this series s_• converges in X to some point of D.

Examples

Although cobal(S) = co(bal(S)), the convex balanced hull of S is not necessarily equal to the balanced hull of the convex hull of S.^[1] For an example where cobal(S) ≠ bal(co(S)), let X be the real vector space ℝ² and let S := { (-1, 1), (1, 1)}. Then bal(co(S)) is a strict subset of cobal(S) 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 cobal(S) is equal to the closed square in X with vertices (-1, 1), (1, 1), (-1, -1), and (-1, 1) while bal(co(S)) is a closed "hour glass shaped" shaped subset that intersects the x-axis at the origin and is the union of two triangles: one whose vertices are the origin along with S and the other triangle whose vertices are the origin along with -S = { (-1, -1), (1, -1)}.