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

A subset S of a real or complex vector space X is called a ' and is said to be ', ', and ' if any of the following equivalent conditions is satisfied:

<ol> <li>S is a [convex](https://bliptext.com/articles/convex-set) and [balanced set](https://bliptext.com/articles/balanced-set).</li> <li>for any scalars a and b, if then </li> <li>for all scalars a, b, and c, if then </li> <li>for any scalars and c, if then </li> <li>for any scalars if then </li> </ol> The smallest [convex](https://bliptext.com/articles/convex-set) (respectively, [balanced](https://bliptext.com/articles/balanced-set)) subset of X containing a given set is called the [convex](https://bliptext.com/articles/convex-set) hull (respectively, the [balanced](https://bliptext.com/articles/balanced-set) hull) of that set and is denoted by (respectively, ). Similarly, the **', the '**, and the of a set S is defined to be the smallest disk (with respect to subset [inclusion](https://bliptext.com/articles/inclusion-mathematics)) containing S. The disked hull of S will be denoted by or and it is equal to each of the following sets: <ol> <li> which is the convex hull of the [balanced hull](https://bliptext.com/articles/balanced-set) of S; thus, <li>the intersection of all disks containing S.</li> <li></li> </ol> # [The light gray area is the absolutely convex hull of the cross. | upload.wikimedia.org/wikipedia/commons/0/0a/Absolute///convex///hull.svg]

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 D is absorbing in X if and only if

Properties

If S is an absorbing disk in a vector space X then there exists an absorbing disk E in X such that If D is a disk and r and s are scalars then s D = |s| D and The absolutely convex hull of a bounded set in a locally convex topological vector space is again bounded. If D is a bounded disk in a TVS X and if is a sequence in D, then the partial sums are Cauchy, where for all n, In particular, if in addition D is a sequentially complete subset of X, then this series s_{\bull} converges in X to some point of D. The convex balanced hull of S contains both the convex hull of S and the balanced hull of S. Furthermore, it contains the balanced hull of the convex hull of S; thus where the example below shows that this inclusion might be strict. However, for any subsets if then which implies

Examples

Although the convex balanced hull of S is necessarily equal to the balanced hull of the convex hull of S. For an example where let X be the real vector space \R^2 and let Then is a strict subset of that is not even convex; in particular, this example also shows that the balanced hull of a convex set is necessarily convex. The set is equal to the closed and filled square in X with vertices and (1, -1) (this is because the balanced set must contain both S and where since is also convex, it must consequently contain the solid square which for this particular example happens to also be balanced so that ). However, is equal to the horizontal closed line segment between the two points in S so that is instead a closed "hour glass shaped" subset that intersects the x-axis at exactly the origin and is the union of two closed and filled isosceles triangles: one whose vertices are the origin together with S and the other triangle whose vertices are the origin together with This non-convex filled "hour-glass" is a proper subset of the filled square

Generalizations

Given a fixed real number a is any subset C of a vector space X with the property that whenever c, d \in C and r, s \geq 0 are non-negative scalars satisfying It is called an or a if whenever c, d \in C and r, s are scalars satisfying A is any non-negative function that satisfies the following conditions: This generalizes the definition of seminorms since a map is a seminorm if and only if it is a 1-seminorm (using p := 1). There exist p-seminorms that are not seminorms. For example, whenever 0 < p < 1 then the map used to define the Lp space L_p(\R) is a p-seminorm but not a seminorm. Given a topological vector space is (meaning that its topology is induced by some p-seminorm) if and only if it has a bounded p-convex neighborhood of the origin.