View source for Absolutely convex set

In [[Mathematics|mathematics]], a [[Subset|subset]] ''C'' of a [[Real number|real]] or [[Complex number|complex]] [[Vector space|vector space]] is said to be '''absolutely convex''' or '''disked''' if it is [[Convex set|convex]] and [[Balanced set|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 (set theory)|intersection]] of all disks containing that set. 

==Definition ==

[[File:Absolute convex hull.svg|thumb|right|The light gray area is the absolutely convex hull of the cross.]]

A subset <math>S</math> of a real or complex vector space <math>X</math> is called a ''{{visible anchor|disk}}'' and is said to be ''{{visible anchor|disked}}'', ''{{visible anchor|absolutely convex}}'', and ''{{visible anchor|convex balanced}}'' if any of the following equivalent conditions is satisfied: 
<ol>
<li><math>S</math> is a [[Convex set|convex]] and [[Balanced set|balanced set]].</li>
<li>for any scalars <math>a</math> and <math>b,</math> if <math>|a| + |b| \leq 1</math> then <math>a S + b S \subseteq S.</math></li>
<li>for all scalars <math>a, b,</math> and <math>c,</math> if <math>|a| + |b| \leq |c|,</math> then <math>a S + b S \subseteq c S.</math></li>
<li>for any scalars <math>a_1, \ldots, a_n</math> and <math>c,</math> if <math>|a_1| + \cdots + |a_n| \leq |c|</math> then <math>a_1 S + \cdots + a_n S \subseteq c S.</math></li>
<li>for any scalars <math>a_1, \ldots, a_n,</math> if <math>|a_1| + \cdots + |a_n| \leq 1</math> then <math>a_1 S + \cdots + a_n S \subseteq S.</math></li>
</ol>

The smallest [[Convex set|convex]] (respectively, [[Balanced set|balanced]]) subset of <math>X</math> containing a given set is called the [[Convex hull|convex hull]] (respectively, the balanced hull) of that set and is denoted by <math>\operatorname{co} S</math> (respectively, <math>\operatorname{bal} S</math>). 

Similarly,  the ''{{visible anchor|disked hull}}'', the ''{{visible anchor|absolute convex hull}}'', and the ''{{visible anchor|convex balanced hull}}'' of a set <math>S</math> is defined to be the smallest disk (with respect to subset inclusion) containing <math>S.</math>{{sfn|Trèves|2006|p=68}}  
The disked hull of <math>S</math> will be denoted by <math>\operatorname{disk} S</math> or <math>\operatorname{cobal} S</math> and it is equal to each of the following sets: 
<ol>
<li><math>\operatorname{co} (\operatorname{bal} S),</math> which is the convex hull of the [[Balanced set|balanced hull]] of <math>S</math>; thus, <math>\operatorname{cobal} S = \operatorname{co} (\operatorname{bal} S).</math>
* In general, <math>\operatorname{cobal} S \neq \operatorname{bal} (\operatorname{co} S)</math> is possible, even in [[Dimension (vector space)|finite dimensional]] vector spaces.</li>
<li>the intersection of all disks containing <math>S.</math></li>
<li><math>\left\{a_1 s_1 + \cdots a_n s_n ~:~ n \in \N, \, s_1, \ldots, s_n \in S, \, \text{ and } a_1, \ldots, a_n \text{ are scalars satisfying } |a_1| + \cdots + |a_n| \leq 1\right\}.</math></li>
</ol>

==Sufficient conditions ==

The intersection of arbitrarily many absolutely convex sets is again absolutely convex; however, [[Union (set theory)|unions]] of absolutely convex sets need not be absolutely convex anymore.

If <math>D</math> is a disk in <math>X,</math> then <math>D</math> is absorbing in <math>X</math> if and only if <math>\operatorname{span} D = X.</math>{{sfn|Narici|Beckenstein|2011|pp=67-113}}

==Properties==


If <math>S</math> is an [[Absorbing set|absorbing]] disk in a vector space <math>X</math> then there exists an absorbing disk <math>E</math> in <math>X</math> such that <math>E + E \subseteq S.</math>{{sfn|Narici|Beckenstein|2011|pp=149-153}} 
If <math>D</math> is a disk and <math>r</math> and <math>s</math> are scalars then <math>s D = |s| D</math> and <math>(r D) \cap (s D) = (\min_{} \{|r|, |s|\}) D.</math> 

The absolutely convex hull of a [[Bounded set (topological vector space)|bounded set]] in a [[Locally convex topological vector space|locally convex topological vector space]] is again bounded.

If <math>D</math> is a bounded disk in a TVS <math>X</math> and if <math>x_{\bull} = \left(x_i\right)_{i=1}^{\infty}</math> is a [[Sequence|sequence]] in <math>D,</math> then the partial sums <math>s_{\bull} = \left(s_n\right)_{n=1}^{\infty}</math> are [[Cauchy sequence|Cauchy]], where for all <math>n,</math> <math>s_n := \sum_{i=1}^n 2^{-i} x_i.</math>{{sfn|Narici|Beckenstein|2011|p=471}} In particular, if in addition <math>D</math> is a [[Sequentially complete|sequentially complete]] subset of <math>X,</math> then this series <math>s_{\bull}</math> converges in <math>X</math> to some point of <math>D.</math> 

The convex balanced hull of <math>S</math> contains both the convex hull of <math>S</math> and the balanced hull of <math>S.</math> Furthermore, it contains the balanced hull of the convex hull of <math>S;</math> thus 
<math display=block>\operatorname{bal} (\operatorname{co} S) ~\subseteq~ \operatorname{cobal} S ~=~ \operatorname{co} (\operatorname{bal} S),</math>
where the example below shows that this inclusion might be strict. 
However, for any subsets <math>S, T \subseteq X,</math> if <math>S \subseteq T</math> then <math>\operatorname{cobal} S \subseteq \operatorname{cobal} T</math>  which implies <math>\operatorname{cobal} (\operatorname{co} S) = \operatorname{cobal} S = \operatorname{cobal} (\operatorname{bal} S).</math> 

===Examples===

Although <math>\operatorname{cobal} S = \operatorname{co} (\operatorname{bal} S),</math> the convex balanced hull of <math>S</math> is {{em|not}} necessarily equal to the balanced hull of the convex hull of <math>S.</math>{{sfn|Trèves|2006|p=68}} 
For an example where <math>\operatorname{cobal} S \neq \operatorname{bal} (\operatorname{co} S)</math> let <math>X</math> be the real vector space <math>\R^2</math> and let <math>S := \{(-1, 1), (1, 1)\}.</math> 
Then <math>\operatorname{bal} (\operatorname{co} S)</math> is a strict subset of <math>\operatorname{cobal} S</math> that is not even convex; 
in particular, this example also shows that the balanced hull of a convex set is {{em|not}} necessarily convex. 
The set <math>\operatorname{cobal} S</math> is equal to the closed and filled square in <math>X</math> with vertices <math>(-1, 1), (1, 1), (-1, -1),</math> and <math>(1, -1)</math> (this is because the balanced set <math>\operatorname{cobal} S</math> must contain both <math>S</math> and <math>-S = \{(-1, -1), (1, -1)\},</math> where since <math>\operatorname{cobal} S</math> is also convex, it must consequently contain the solid square <math>\operatorname{co} ((-S) \cup S),</math> which for this particular example happens to also be balanced so that <math>\operatorname{cobal} S = \operatorname{co} ((-S) \cup S)</math>). However, <math>\operatorname{co} (S)</math> is equal to the horizontal closed line segment between the two points in <math>S</math> so that <math>\operatorname{bal} (\operatorname{co} S)</math> is instead a closed "hour glass shaped" subset that intersects the <math>x</math>-axis at exactly the origin and is the union of two closed and filled [[Isosceles triangle|isosceles triangle]]s: one whose vertices are the origin together with <math>S</math> and the other triangle whose vertices are the origin together with <math>- S = \{(-1, -1), (1, -1)\}.</math> This non-convex filled "hour-glass" <math>\operatorname{bal} (\operatorname{co} S)</math> is a proper subset of the filled square <math>\operatorname{cobal} S = \operatorname{co} (\operatorname{bal} S).</math>

==Generalizations==

Given a fixed real number <math>0 < p \leq 1,</math> a {{em|{{visible anchor|p-convex set|text=<math>p</math>-convex set}}}} is any subset <math>C</math> of a vector space <math>X</math> with the property that <math>r c + s d \in C</math> whenever <math>c, d \in C</math> and <math>r, s \geq 0</math> are non-negative scalars satisfying <math>r^p + s^p = 1.</math> 
It is called an {{em|{{visible anchor|absolutely p-convex set|text=absolutely <math>p</math>-convex set}}}} or a {{em|{{visible anchor|p-disk|text=<math>p</math>-disk}}}} if <math>r c + s d \in C</math> whenever <math>c, d \in C</math> and <math>r, s</math> are scalars satisfying <math>|r|^p + |s|^p \leq 1.</math>{{sfn|Narici|Beckenstein|2011|p=174}} 

A {{em|{{visible anchor|p-seminorm|text=<math>p</math>-seminorm}}}}{{sfn|Narici|Beckenstein|2011|p=86}} is any non-negative function <math>q : X \to \R</math> that satisfies the following conditions:
# Subadditivity/[[Triangle inequality]]: <math>q(x + y) \leq q(x) + q(y)</math> for all <math>x, y \in X.</math>
# [[Homogeneous function#Absolute homogeneity of degree|Absolute homogeneity of degree <math>p</math>]]: <math>q(s x) =|s|^p q(x)</math> for all <math>x \in X</math> and all scalars <math>s.</math>

This generalizes the definition of [[Seminorm|seminorm]]s since a map is a seminorm if and only if it is a <math>1</math>-seminorm (using <math>p := 1</math>). 
There exist <math>p</math>-seminorms that are not [[Seminorm|seminorm]]s. For example, whenever <math>0 < p < 1</math> then the map <math>q(f) = \int_{\R} |f(t)|^p d t</math> used to define the [[Lp space]] <math>L_p(\R)</math> is a <math>p</math>-seminorm but not a seminorm.{{sfn|Narici|Beckenstein|2011|p=86}}

Given <math>0 < p \leq 1,</math> a [[Topological vector space|topological vector space]] is {{em|<math>p</math>-seminormable}} (meaning that its topology is induced by some <math>p</math>-seminorm) if and only if it has a [[Bounded set (topological vector space)|bounded]] <math>p</math>-convex neighborhood of the origin.{{sfn|Narici|Beckenstein|2011|p=174}}

==See also==


* {{annotated link|Absorbing set}}
* {{annotated link|Auxiliary normed space}}
* {{annotated link|Balanced set}}
* {{annotated link|Bounded set (topological vector space)}}
* {{annotated link|Convex set}}
* {{annotated link|Star domain}}
* {{annotated link|Symmetric set}}
* {{annotated link|Vector field}}

==References==

{{reflist}}
{{reflist|group=note}}

==Bibliography==

* {{cite book|last=Robertson|first=A.P.|author2=W.J. Robertson|title=Topological vector spaces|series=Cambridge Tracts in Mathematics|volume=53|year=1964|publisher={{wipe|Cambridge University Press}}|pages=4–6}}
* {{Narici Beckenstein Topological Vector Spaces|edition=2}} <!--{{sfn|Narici|Beckenstein|2011|p=}}-->
* {{Schaefer Wolff Topological Vector Spaces|edition=2}} <!--{{sfn|Schaefer|Wolff|1999|p=}}-->
* {{Trèves François Topological vector spaces, distributions and kernels}} <!--{{sfn|Trèves|2006|p=}}-->

{{Functional analysis}}

[[Category:Abstract algebra]]
[[Category:Convex analysis]]
[[Category:Convex geometry]]
[[Category:Group theory]]
[[Category:Linear algebra]]

{{Sourceattribution|Absolutely convex set}}