absolutely convex set

File:Absolute convex hull.svg

A subset $S$ of a real or complex vector space $X$ 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:

1. $S$ is a convex and balanced set.
2. for any scalars $a$ and $b,$ if $|a|\; +\; |b|\; \backslash leq\; 1$ then $a\; S\; +\; b\; S\; \backslash subseteq\; S.$
3. for all scalars $a,\; b,$ and $c,$ if $|a|\; +\; |b|\; \backslash leq\; |c|,$ then $a\; S\; +\; b\; S\; \backslash subseteq\; c\; S.$
4. for any scalars $a\_1,\; \backslash ldots,\; a\_n$ and $c,$ if $|a\_1|\; +\; \backslash cdots\; +\; |a\_n|\; \backslash leq\; |c|$ then $a\_1\; S\; +\; \backslash cdots\; +\; a\_n\; S\; \backslash subseteq\; c\; S.$
5. for any scalars $a\_1,\; \backslash ldots,\; a\_n,$ if $|a\_1|\; +\; \backslash cdots\; +\; |a\_n|\; \backslash leq\; 1$ then $a\_1\; S\; +\; \backslash cdots\; +\; a\_n\; S\; \backslash subseteq\; S.$

The smallest convex (respectively, balanced) subset of $X$ containing a given set is called the convex hull (respectively, the balanced hull) of that set and is denoted by $\backslash operatorname\{co\}\; S$ (respectively, $\backslash operatorname\{bal\}\; S$).

Similarly, the {{visible anchor|disked hull}}, the {{visible anchor|absolute convex hull}}, and the {{visible anchor|convex balanced hull}} of a set $S$ is defined to be the smallest disk (with respect to subset inclusion) containing $S.${{sfn|Trèves|2006|p=68}}

The disked hull of $S$ will be denoted by $\backslash operatorname\{disk\}\; S$ or $\backslash operatorname\{cobal\}\; S$ and it is equal to each of the following sets:

1. $\backslash operatorname\{co\}\; (\backslash operatorname\{bal\}\; S),$ which is the convex hull of the balanced hull of $S$; thus, $\backslash operatorname\{cobal\}\; S\; =\; \backslash operatorname\{co\}\; (\backslash operatorname\{bal\}\; S).$
   • In general, $\backslash operatorname\{cobal\}\; S\; \backslash neq\; \backslash operatorname\{bal\}\; (\backslash operatorname\{co\}\; S)$ is possible, even in finite dimensional vector spaces.

2. the intersection of all disks containing $S.$
3. $\backslash left\backslash \{a\_1\; s\_1\; +\; \backslash cdots\; a\_n\; s\_n\; ~:~\; n\; \backslash in\; \backslash N,\; \backslash ,\; s\_1,\; \backslash ldots,\; s\_n\; \backslash in\; S,\; \backslash ,\; \backslash text\{\; and\; \}\; a\_1,\; \backslash ldots,\; a\_n\; \backslash text\{\; are\; scalars\; satisfying\; \}\; |a\_1|\; +\; \backslash cdots\; +\; |a\_n|\; \backslash leq\; 1\backslash right\backslash \}.$

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 $\backslash operatorname\{span\}\; D\; =\; X.${{sfn|Narici|Beckenstein|2011|pp=67-113}}

{{See also|Topological vector space#Properties}}

If $S$ is an absorbing disk in a vector space $X$ then there exists an absorbing disk $E$ in $X$ such that $E\; +\; E\; \backslash subseteq\; S.${{sfn|Narici|Beckenstein|2011|pp=149-153}}

If $D$ is a disk and $r$ and $s$ are scalars then $s\; D\; =\; |s|\; D$ and $(r\; D)\; \backslash cap\; (s\; D)\; =\; (\backslash min\_\{\}\; \backslash \{|r|,\; |s|\backslash \})\; D.$

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 $x\_\{\backslash bull\}\; =\; \backslash left(x\_i\backslash right)\_\{i=1\}^\{\backslash infty\}$ is a sequence in $D,$ then the partial sums $s\_\{\backslash bull\}\; =\; \backslash left(s\_n\backslash right)\_\{n=1\}^\{\backslash infty\}$ are Cauchy, where for all $n,$ $s\_n\; :=\; \backslash sum\_\{i=1\}^n\; 2^\{-i\}\; x\_i.${{sfn|Narici|Beckenstein|2011|p=471}} In particular, if in addition $D$ is a sequentially complete subset of $X,$ then this series $s\_\{\backslash bull\}$ 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

$$\backslash operatorname\{bal\}\; (\backslash operatorname\{co\}\; S)\; ~\backslash subseteq~\; \backslash operatorname\{cobal\}\; S\; ~=~\; \backslash operatorname\{co\}\; (\backslash operatorname\{bal\}\; S),$$

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

However, for any subsets $S,\; T\; \backslash subseteq\; X,$ if $S\; \backslash subseteq\; T$ then $\backslash operatorname\{cobal\}\; S\; \backslash subseteq\; \backslash operatorname\{cobal\}\; T$ which implies $\backslash operatorname\{cobal\}\; (\backslash operatorname\{co\}\; S)\; =\; \backslash operatorname\{cobal\}\; S\; =\; \backslash operatorname\{cobal\}\; (\backslash operatorname\{bal\}\; S).$

Although $\backslash operatorname\{cobal\}\; S\; =\; \backslash operatorname\{co\}\; (\backslash operatorname\{bal\}\; S),$ the convex balanced hull of $S$ is {{em|not}} necessarily equal to the balanced hull of the convex hull of $S.${{sfn|Trèves|2006|p=68}}

For an example where $\backslash operatorname\{cobal\}\; S\; \backslash neq\; \backslash operatorname\{bal\}\; (\backslash operatorname\{co\}\; S)$ let $X$ be the real vector space $\backslash R^2$ and let $S\; :=\; \backslash \{(-1,\; 1),\; (1,\; 1)\backslash \}.$

Then $\backslash operatorname\{bal\}\; (\backslash operatorname\{co\}\; S)$ is a strict subset of $\backslash operatorname\{cobal\}\; S$ 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 $\backslash operatorname\{cobal\}\; S$ is equal to the closed and filled square in $X$ with vertices $(-1,\; 1),\; (1,\; 1),\; (-1,\; -1),$ and $(1,\; -1)$ (this is because the balanced set $\backslash operatorname\{cobal\}\; S$ must contain both $S$ and $-S\; =\; \backslash \{(-1,\; -1),\; (1,\; -1)\backslash \},$ where since $\backslash operatorname\{cobal\}\; S$ is also convex, it must consequently contain the solid square $\backslash operatorname\{co\}\; ((-S)\; \backslash cup\; S),$ which for this particular example happens to also be balanced so that $\backslash operatorname\{cobal\}\; S\; =\; \backslash operatorname\{co\}\; ((-S)\; \backslash cup\; S)$). However, $\backslash operatorname\{co\}\; (S)$ is equal to the horizontal closed line segment between the two points in $S$ so that $\backslash operatorname\{bal\}\; (\backslash operatorname\{co\}\; S)$ 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 $-\; S\; =\; \backslash \{(-1,\; -1),\; (1,\; -1)\backslash \}.$ This non-convex filled "hour-glass" $\backslash operatorname\{bal\}\; (\backslash operatorname\{co\}\; S)$ is a proper subset of the filled square $\backslash operatorname\{cobal\}\; S\; =\; \backslash operatorname\{co\}\; (\backslash operatorname\{bal\}\; S).$

Given a fixed real number a {{em|{{visible anchor|p-convex set|text=$p$-convex set}}}} is any subset $C$ of a vector space $X$ with the property that $r\; c\; +\; s\; d\; \backslash in\; C$ whenever $c,\; d\; \backslash in\; C$ and $r,\; s\; \backslash geq\; 0$ are non-negative scalars satisfying $r^p\; +\; s^p\; =\; 1.$

It is called an {{em|{{visible anchor|absolutely p-convex set|text=absolutely $p$-convex set}}}} or a {{em|{{visible anchor|p-disk|text=$p$-disk}}}} if $r\; c\; +\; s\; d\; \backslash in\; C$ whenever $c,\; d\; \backslash in\; C$ and $r,\; s$ are scalars satisfying $|r|^p\; +\; |s|^p\; \backslash leq\; 1.${{sfn|Narici|Beckenstein|2011|p=174}}

A {{em|{{visible anchor|p-seminorm|text=$p$-seminorm}}}}{{sfn|Narici|Beckenstein|2011|p=86}} is any non-negative function $q\; :\; X\; \backslash to\; \backslash R$ that satisfies the following conditions:

1. Subadditivity/Triangle inequality: $q(x\; +\; y)\; \backslash leq\; q(x)\; +\; q(y)$ for all $x,\; y\; \backslash in\; X.$
2. Absolute homogeneity of degree $p$: $q(s\; x)\; =|s|^p\; q(x)$ for all $x\; \backslash in\; X$ and all scalars $s.$

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 then the map $q(f)\; =\; \backslash int\_\{\backslash R\}\; |f(t)|^p\; d\; t$ used to define the Lp space $L\_p(\backslash R)$ is a $p$-seminorm but not a seminorm.{{sfn|Narici|Beckenstein|2011|p=86}}

Given a topological vector space is {{em|$p$-seminormable}} (meaning that its topology is induced by some $p$-seminorm) if and only if it has a bounded $p$-convex neighborhood of the origin.{{sfn|Narici|Beckenstein|2011|p=174}}

{{Wikibooks|Algebra|Vector spaces}}

• {{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 (geometric)}}, for vectors in physics
• {{annotated link|Vector field}}

{{reflist}}

{{reflist|group=note}}

• {{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=Cambridge University Press|pages=4–6}}
• {{Narici Beckenstein Topological Vector Spaces|edition=2}}
• {{Schaefer Wolff Topological Vector Spaces|edition=2}}
• {{Trèves François Topological vector spaces, distributions and kernels}}

{{Functional analysis}}

{{Topological vector spaces}}

{{Convex analysis and variational analysis}}

Category:Abstract algebra

Category:Convex analysis

Category:Convex geometry

Category:Group theory

Category:Linear algebra