Absolutely convex set

Template:Short description 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

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

A subset $S$ of a real or complex vector space $X$ is called a Template:Visible anchor and is said to be Template:Visible anchor, Template:Visible anchor, and Template:Visible anchor 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 $|a|+|b|\le 1$ then $aS+bS\subseteq S.$
3. for all scalars $a,b,$ and $c,$ if $|a|+|b|\le |c|,$ then $aS+bS\subseteq cS.$
4. for any scalars $a_1,\dots ,a_n$ and $c,$ if $|a_1|+\cdots +|a_n|\le |c|$ then $a_{1}S+\cdots +a_{n}S\subseteq cS.$
5. for any scalars $a_1,\dots ,a_n,$ if $|a_1|+\cdots +|a_n|\le 1$ then $a_{1}S+\cdots +a_{n}S\subseteq S.$

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 $\mathrm{co}S$ (respectively, $\mathrm{bal}S$).

Similarly, the Template:Visible anchor, the Template:Visible anchor, and the Template:Visible anchor of a set $S$ is defined to be the smallest disk (with respect to subset inclusion) containing $S.$Template:Sfn The disked hull of $S$ will be denoted by $\mathrm{disk}S$ or $\mathrm{cobal}S$ and it is equal to each of the following sets:

1. $\mathrm{co}(\mathrm{bal}S),$ which is the convex hull of the balanced hull of $S$; thus, $\mathrm{cobal}S=\mathrm{co}(\mathrm{bal}S).$
   • In general, $\mathrm{cobal}S\ne \mathrm{bal}(\mathrm{co}S)$ is possible, even in finite dimensional vector spaces.
2. the intersection of all disks containing $S.$
3. $\{a_1{s}_1+\cdots a_n{s}_n:n\in \mathbb{N},s_1,\dots ,s_n\in S,\text{ and }{a}_1,\dots ,a_n\text{ are scalars satisfying }|a_1|+\cdots +|a_n|\le 1\}.$

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 $D$ is a disk in $X,$ then $D$ is absorbing in $X$ if and only if $\mathrm{span}D=X.$Template:Sfn

Properties

If $S$ is an absorbing disk in a vector space $X$ then there exists an absorbing disk $E$ in $X$ such that $E+E\subseteq S.$Template:Sfn If $D$ is a disk and $r$ and $s$ are scalars then $sD=|s|D$ and $(rD)\cap (sD)=(\underset{}{\min }\{|r|,|s|\})D.$

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 $x_{\bullet }={\left(x_i\right)}_{i=1}^{\mathrm{\infty }}$ is a sequence in $D,$ then the partial sums $s_{\bullet }={\left(s_n\right)}_{n=1}^{\mathrm{\infty }}$ are Cauchy, where for all $n,$ $s_n:=\sum _{i=1}^n{2}^{-i}{x}_i.$Template:Sfn In particular, if in addition $D$ is a sequentially complete subset of $X,$ then this series $s_{\bullet }$ 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 $$\mathrm{bal}(\mathrm{co}S)\subseteq \mathrm{cobal}S=\mathrm{co}(\mathrm{bal}S),$$ where the example below shows that this inclusion might be strict. However, for any subsets $S,T\subseteq X,$ if $S\subseteq T$ then $\mathrm{cobal}S\subseteq \mathrm{cobal}T$ which implies $\mathrm{cobal}(\mathrm{co}S)=\mathrm{cobal}S=\mathrm{cobal}(\mathrm{bal}S).$

Examples

Although $\mathrm{cobal}S=\mathrm{co}(\mathrm{bal}S),$ the convex balanced hull of $S$ is Template:Em necessarily equal to the balanced hull of the convex hull of $S.$Template:Sfn For an example where $\mathrm{cobal}S\ne \mathrm{bal}(\mathrm{co}S)$ let $X$ be the real vector space ${ℝ}^2$ and let $S:=\{(-1,1),(1,1)\}.$ Then $\mathrm{bal}(\mathrm{co}S)$ is a strict subset of $\mathrm{cobal}S$ that is not even convex; in particular, this example also shows that the balanced hull of a convex set is Template:Em necessarily convex. The set $\mathrm{cobal}S$ 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 $\mathrm{cobal}S$ must contain both $S$ and $-S=\{(-1,-1),(1,-1)\},$ where since $\mathrm{cobal}S$ is also convex, it must consequently contain the solid square $\mathrm{co}((-S)\cup S),$ which for this particular example happens to also be balanced so that $\mathrm{cobal}S=\mathrm{co}((-S)\cup S)$). However, $\mathrm{co}(S)$ is equal to the horizontal closed line segment between the two points in $S$ so that $\mathrm{bal}(\mathrm{co}S)$ 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 $-S=\{(-1,-1),(1,-1)\}.$ This non-convex filled "hour-glass" $\mathrm{bal}(\mathrm{co}S)$ is a proper subset of the filled square $\mathrm{cobal}S=\mathrm{co}(\mathrm{bal}S).$

Generalizations

Given a fixed real number $0<p\le 1,$ a Template:Em is any subset $C$ of a vector space $X$ with the property that $rc+sd\in C$ whenever $c,d\in C$ and $r,s\ge 0$ are non-negative scalars satisfying $r^p+s^p=1.$ It is called an Template:Em or a Template:Em if $rc+sd\in C$ whenever $c,d\in C$ and $r,s$ are scalars satisfying $|r{|}^p+|s{|}^p\le 1.$Template:Sfn

A Template:EmTemplate:Sfn is any non-negative function $q:X\to ℝ$ that satisfies the following conditions:

1. Subadditivity/Triangle inequality: $q(x+y)\le q(x)+q(y)$ for all $x,y\in X.$
2. Absolute homogeneity of degree $p$: $q(sx)=|s{|}^{p}q(x)$ for all $x\in X$ 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 $0<p<1$ then the map $q(f)=\underset{ℝ}{\int }|f(t){|}^{p}dt$ used to define the Lp space $L_p(ℝ)$ is a $p$-seminorm but not a seminorm.Template:Sfn

Given $0<p\le 1,$ a topological vector space is Template:Em (meaning that its topology is induced by some $p$-seminorm) if and only if it has a bounded $p$-convex neighborhood of the origin.Template:Sfn

See also

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Bibliography

• Template:Cite book
• Template:Narici Beckenstein Topological Vector Spaces
• Template:Schaefer Wolff Topological Vector Spaces
• Template:Trèves François Topological vector spaces, distributions and kernels