Absolutely convex set

Summary

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

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

A subset   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:

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

Similarly, the disked hull, the absolute convex hull, and the convex balanced hull of a set   is defined to be the smallest disk (with respect to subset inclusion) containing  [1] 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,  
    • In general,   is possible, even in finite dimensional vector spaces.
  2. the intersection of all disks containing  
  3.  

Sufficient conditions

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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  [2]

Properties

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If   is an absorbing disk in a vector space   then there exists an absorbing disk   in   such that  [3] 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    [4] 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

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Although   the convex balanced hull of   is not necessarily equal to the balanced hull of the convex hull of  [1] 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 not necessarily convex. The set   is equal to the closed and filled square in   with vertices   and   (this is because the balanced set   must contain both   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   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

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Given a fixed real number   a  -convex set is any subset   of a vector space   with the property that   whenever   and   are non-negative scalars satisfying   It is called an absolutely  -convex set or a  -disk if   whenever   and   are scalars satisfying  [5]

A  -seminorm[6] 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 (using  ). 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.[6]

Given   a topological vector space is  -seminormable (meaning that its topology is induced by some  -seminorm) if and only if it has a bounded  -convex neighborhood of the origin.[5]

See also

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References

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Bibliography

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  • Robertson, A.P.; W.J. Robertson (1964). Topological vector spaces. Cambridge Tracts in Mathematics. Vol. 53. Cambridge University Press. pp. 4–6.
  • Narici, Lawrence; Beckenstein, Edward (2011). Topological Vector Spaces. Pure and applied mathematics (Second ed.). Boca Raton, FL: CRC Press. ISBN 978-1584888666. OCLC 144216834.
  • Schaefer, Helmut H.; Wolff, Manfred P. (1999). Topological Vector Spaces. GTM. Vol. 8 (Second ed.). New York, NY: Springer New York Imprint Springer. ISBN 978-1-4612-7155-0. OCLC 840278135.
  • Trèves, François (2006) [1967]. Topological Vector Spaces, Distributions and Kernels. Mineola, N.Y.: Dover Publications. ISBN 978-0-486-45352-1. OCLC 853623322.