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.

of a real or complex vector space

is called a disk and is said to be disked, absolutely convex, and convex balanced if any of the following equivalent conditions is satisfied: The smallest convex (respectively, balanced) subset of

containing a given set is called the convex hull (respectively, the balanced hull) of that set and is denoted by

is defined to be the smallest disk (with respect to subset inclusion) containing

and it is equal to each of the following sets: The intersection of arbitrarily many absolutely convex sets is again absolutely convex; however, unions of absolutely convex sets need not be absolutely convex anymore.

is an absorbing disk in a vector space

then there exists an absorbing disk

The absolutely convex hull of a bounded set in a locally convex topological vector space is again bounded.

is a bounded disk in a TVS

is a sequentially complete subset of

The convex balanced hull of

where the example below shows that this inclusion might be strict.

the convex balanced hull of

be the real vector space

is equal to the closed and filled square in

is also convex, it must consequently contain the solid square

is equal to the horizontal closed line segment between the two points in

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

Given a fixed real number

are non-negative scalars satisfying

that satisfies the following conditions: This generalizes the definition of seminorms since a map is a seminorm if and only if it is a

used to define the Lp space

a topological vector space is

-seminormable (meaning that its topology is induced by some