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

  1. is a convex and balanced set.
  2. for any scalars and if then
  3. for all scalars and if then
  4. for any scalars and if then
  5. for any scalars if then
The smallest convex subset of containing a given set is called the convex hull of that set and is denoted by .
Similarly, the ', the ', and the 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,
  2. the intersection of all disks containing

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 is a disk in then is absorbing in if and only if

Properties

If is an absorbing disk in a vector space then there exists an absorbing disk in such that
If is a disk and and are scalars then and
The absolutely convex hull of a bounded set in a locally convex topological vector space is again bounded.
If is a bounded disk in a TVS and if is a sequence in then the partial sums are Cauchy, where for all In particular, if in addition is a sequentially complete subset of then this series converges in to some point of
The convex balanced hull of contains both the convex hull of and the balanced hull of Furthermore, it contains the balanced hull of the convex hull of 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 is 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 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 with vertices and . However, is equal to the horizontal closed line segment between the two points in so that is instead a closed "hour glass shaped" subset that intersects the -axis at exactly the origin and is the union of two closed and filled isosceles triangles: one whose vertices are the origin together with 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 of a vector space with the property that whenever and are non-negative scalars satisfying
It is called an or a if whenever and are scalars satisfying
A is any non-negative function that satisfies the following conditions:
  1. Subadditivity/Triangle inequality: for all
  2. Absolute homogeneity of degree : for all and all scalars
This generalizes the definition of seminorms since a map is a seminorm if and only if it is a -seminorm.
There exist -seminorms that are not seminorms. For example, whenever then the map used to define the Lp space is a -seminorm but not a seminorm.
Given a topological vector space is if and only if it has a bounded -convex neighborhood of the origin.