Absolutely convex set

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

A subset ${\displaystyle S}$ of a real or complex vector space ${\displaystyle X}$ 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. ${\displaystyle S}$ is a convex and balanced set.
2. for any scalars ${\displaystyle a}$ and ${\displaystyle b,}$ if ${\displaystyle |a|+|b|\leq 1}$ then ${\displaystyle aS+bS\subseteq S.}$
3. for all scalars ${\displaystyle a,b,}$ and ${\displaystyle c,}$ if ${\displaystyle |a|+|b|\leq |c|,}$ then ${\displaystyle aS+bS\subseteq cS.}$
4. for any scalars ${\displaystyle a_{1},\ldots ,a_{n}}$ and ${\displaystyle c,}$ if ${\displaystyle |a_{1}|+\cdots +|a_{n}|\leq |c|}$ then ${\displaystyle a_{1}S+\cdots +a_{n}S\subseteq cS.}$
5. for any scalars ${\displaystyle a_{1},\ldots ,a_{n},}$ if ${\displaystyle |a_{1}|+\cdots +|a_{n}|\leq 1}$ then ${\displaystyle a_{1}S+\cdots +a_{n}S\subseteq S.}$

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

Similarly, the disked hull, the absolute convex hull, and the convex balanced hull of a set ${\displaystyle S}$ is defined to be the smallest disk (with respect to subset inclusion) containing ${\displaystyle S.}$[1] The disked hull of ${\displaystyle S}$ will be denoted by ${\displaystyle \operatorname {disk} S}$ or ${\displaystyle \operatorname {cobal} S}$ and it is equal to each of the following sets:

1. ${\displaystyle \operatorname {co} (\operatorname {bal} S),}$ which is the convex hull of the balanced hull of ${\displaystyle S}$; thus, ${\displaystyle \operatorname {cobal} S=\operatorname {co} (\operatorname {bal} S).}$
   • In general, ${\displaystyle \operatorname {cobal} S\neq \operatorname {bal} (\operatorname {co} S)}$ is possible, even in finite dimensional vector spaces.
2. the intersection of all disks containing ${\displaystyle S.}$
3. ${\displaystyle \left\{a_{1}s_{1}+\cdots a_{n}s_{n}~:~n\in \mathbb {N} ,\,s_{1},\ldots ,s_{n}\in S,\,{\text{ and }}a_{1},\ldots ,a_{n}{\text{ are scalars satisfying }}|a_{1}|+\cdots +|a_{n}|\leq 1\right\}.}$

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 ${\displaystyle D}$ is a disk in ${\displaystyle X,}$ then ${\displaystyle D}$ is absorbing in ${\displaystyle X}$ if and only if ${\displaystyle \operatorname {span} D=X.}$[2]

If ${\displaystyle S}$ is an absorbing disk in a vector space ${\displaystyle X}$ then there exists an absorbing disk ${\displaystyle E}$ in ${\displaystyle X}$ such that ${\displaystyle E+E\subseteq S.}$[3] If ${\displaystyle D}$ is a disk and ${\displaystyle r}$ and ${\displaystyle s}$ are scalars then ${\displaystyle sD=|s|D}$ and ${\displaystyle (rD)\cap (sD)=(\min _{}\{|r|,|s|\})D.}$

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

If ${\displaystyle D}$ is a bounded disk in a TVS ${\displaystyle X}$ and if ${\displaystyle x_{\bullet }=\left(x_{i}\right)_{i=1}^{\infty }}$ is a sequence in ${\displaystyle D,}$ then the partial sums ${\displaystyle s_{\bullet }=\left(s_{n}\right)_{n=1}^{\infty }}$ are Cauchy, where for all ${\displaystyle n,}$ ${\displaystyle s_{n}:=\sum _{i=1}^{n}2^{-i}x_{i}.}$[4] In particular, if in addition ${\displaystyle D}$ is a sequentially complete subset of ${\displaystyle X,}$ then this series ${\displaystyle s_{\bullet }}$ converges in ${\displaystyle X}$ to some point of ${\displaystyle D.}$

The convex balanced hull of ${\displaystyle S}$ contains both the convex hull of ${\displaystyle S}$ and the balanced hull of ${\displaystyle S.}$ Furthermore, it contains the balanced hull of the convex hull of ${\displaystyle S;}$ thus

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

where the example below shows that this inclusion might be strict. However, for any subsets ${\displaystyle S,T\subseteq X,}$ if ${\displaystyle S\subseteq T}$ then ${\displaystyle \operatorname {cobal} S\subseteq \operatorname {cobal} T}$ which implies ${\displaystyle \operatorname {cobal} (\operatorname {co} S)=\operatorname {cobal} S=\operatorname {cobal} (\operatorname {bal} S).}$

Given a fixed real number ${\displaystyle 0<p\leq 1,}$ a ${\displaystyle p}$-convex set is any subset ${\displaystyle C}$ of a vector space ${\displaystyle X}$ with the property that ${\displaystyle rc+sd\in C}$ whenever ${\displaystyle c,d\in C}$ and ${\displaystyle r,s\geq 0}$ are non-negative scalars satisfying ${\displaystyle r^{p}+s^{p}=1.}$ It is called an absolutely ${\displaystyle p}$-convex set or a ${\displaystyle p}$-disk if ${\displaystyle rc+sd\in C}$ whenever ${\displaystyle c,d\in C}$ and ${\displaystyle r,s}$ are scalars satisfying ${\displaystyle |r|^{p}+|s|^{p}\leq 1.}$[5]

A ${\displaystyle p}$-seminorm[6] is any non-negative function ${\displaystyle q:X\to \mathbb {R} }$ that satisfies the following conditions:

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

This generalizes the definition of seminorms since a map is a seminorm if and only if it is a ${\displaystyle 1}$-seminorm (using ${\displaystyle p:=1}$). There exist ${\displaystyle p}$-seminorms that are not seminorms. For example, whenever ${\displaystyle 0<p<1}$ then the map ${\displaystyle q(f)=\int _{\mathbb {R} }|f(t)|^{p}dt}$ used to define the Lp space ${\displaystyle L_{p}(\mathbb {R} )}$ is a ${\displaystyle p}$-seminorm but not a seminorm.[6]

Given ${\displaystyle 0<p\leq 1,}$ a topological vector space is ${\displaystyle p}$-seminormable (meaning that its topology is induced by some ${\displaystyle p}$-seminorm) if and only if it has a bounded ${\displaystyle p}$-convex neighborhood of the origin.[5]

The Wikibook Algebra has a page on the topic of: Vector spaces

• Absorbing set – Set that can be "inflated" to reach any point
• Auxiliary normed space
• Balanced set – Construct in functional analysis
• Bounded set (topological vector space) – Generalization of boundedness
• Convex set – In geometry, set whose intersection with every line is a single line segment
• Star domain – Property of point sets in Euclidean spaces
• Symmetric set – Property of group subsets (mathematics)
• Vector (geometric) – Geometric object that has length and directionPages displaying short descriptions of redirect targets, for vectors in physics
• Vector field – Assignment of a vector to each point in a subset of Euclidean space

• 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.