Absolutely convex set

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 hullof${\displaystyle S}$ ; 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 sequencein${\displaystyle D,}$  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 ${\displaystyle \operatorname {cobal} (\operatorname {co} S)=\operatorname {cobal} S=\operatorname {cobal} (\operatorname {bal} S).}$ 

Although ${\displaystyle \operatorname {cobal} S=\operatorname {co} (\operatorname {bal} S),}$  the convex balanced hull of ${\displaystyle S}$ isnot necessarily equal to the balanced hull of the convex hull of ${\displaystyle S.}$ ^[1] For an example where ${\displaystyle \operatorname {cobal} S\neq \operatorname {bal} (\operatorname {co} S)}$  let ${\displaystyle X}$  be the real vector space ${\displaystyle \mathbb {R} ^{2}}$  and let ${\displaystyle S:=\{(-1,1),(1,1)\}.}$  Then ${\displaystyle \operatorname {bal} (\operatorname {co} S)}$  is a strict subset of ${\displaystyle \operatorname {cobal} S}$  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 ${\displaystyle \operatorname {cobal} S}$  is equal to the closed and filled square in ${\displaystyle X}$  with vertices ${\displaystyle (-1,1),(1,1),(-1,-1),}$  and ${\displaystyle (1,-1)}$  (this is because the balanced set ${\displaystyle \operatorname {cobal} S}$  must contain both ${\displaystyle S}$  and ${\displaystyle -S=\{(-1,-1),(1,-1)\},}$  where since ${\displaystyle \operatorname {cobal} S}$  is also convex, it must consequently contain the solid square ${\displaystyle \operatorname {co} ((-S)\cup S),}$  which for this particular example happens to also be balanced so that ${\displaystyle \operatorname {cobal} S=\operatorname {co} ((-S)\cup S)}$ ). However, ${\displaystyle \operatorname {co} ($S)}   is equal to the horizontal closed line segment between the two points in ${\displaystyle S}$  so that ${\displaystyle \operatorname {bal} (\operatorname {co} S)}$  is instead a closed "hour glass shaped" subset that intersects the ${\displaystyle x}$ -axis at exactly the origin and is the union of two closed and filled isosceles triangles: one whose vertices are the origin together with ${\displaystyle S}$  and the other triangle whose vertices are the origin together with ${\displaystyle -S=\{(-1,-1),(1,-1)\}.}$  This non-convex filled "hour-glass" ${\displaystyle \operatorname {bal} (\operatorname {co} S)}$  is a proper subset of the filled square ${\displaystyle \operatorname {cobal} S=\operatorname {co} (\operatorname {bal} S).}$ 

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 ${\displaystyle p}$ -diskif${\displaystyle rc+sd\in C}$  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 ${\displaystyle 0<p\leq 1,}$ atopological vector spaceis${\displaystyle p}$ -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]

• 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.