Inmathematics, asubsetC of areal orcomplexvector space is said to beabsolutely convex ordisked if it isconvex andbalanced (some people use the term "circled" instead of "balanced"), in which case it is called adisk. Thedisked hull or theabsolute convex hull of a set is theintersection of all disks containing that set.

A subset${\displaystyle S}$ of a real or complex vector space${\displaystyle X}$ is called adisk and is said to bedisked,absolutely convex, andconvex balanced if any of the following equivalent conditions is satisfied:

1. ${\displaystyle S}$ is aconvex andbalanced set.
2. for any scalars${\displaystyle a}$ and${\displaystyle b,}$ if${\displaystyle |a|+|b|\le 1}$ then${\displaystyle aS+bS\subseteq S.}$
3. for all scalars${\displaystyle a,b,}$ and${\displaystyle c,}$ if${\displaystyle |a|+|b|\le |c|,}$ then${\displaystyle aS+bS\subseteq cS.}$
4. for any scalars${\displaystyle a_1,\dots ,a_n}$ and${\displaystyle c,}$ if${\displaystyle |a_1|+\cdots +|a_n|\le |c|}$ then${\displaystyle a_{1}S+\cdots +a_{n}S\subseteq cS.}$
5. for any scalars${\displaystyle a_1,\dots ,a_n,}$ if${\displaystyle |a_1|+\cdots +|a_n|\le 1}$ then${\displaystyle a_{1}S+\cdots +a_{n}S\subseteq S.}$

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

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

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

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${\displaystyle \mathrm{span}D=X.}$^[2]

If${\displaystyle S}$ is anabsorbing 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${\displaystyle (rD)\cap (sD)=(\underset{}{min}\{|r|,|s|\})D.}$

The absolutely convex hull of abounded set in alocally convex topological vector space is again bounded.

If${\displaystyle D}$ is a bounded disk in a TVS${\displaystyle X}$ and if${\displaystyle x_{\bullet }={\left(x_i\right)}_{i=1}^{\mathrm{\infty }}}$ is asequence in${\displaystyle D,}$ then the partial sums${\displaystyle s_{\bullet }={\left(s_n\right)}_{n=1}^{\mathrm{\infty }}}$ areCauchy, 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 asequentially 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$${\displaystyle \mathrm{bal}(\mathrm{co}S)\subseteq \mathrm{cobal}S=\mathrm{co}(\mathrm{bal}S),}$$where the example below shows that this inclusion might be strict. However, for any subsets${\displaystyle S,T\subseteq X,}$ if${\displaystyle S\subseteq T}$ then${\displaystyle \mathrm{cobal}S\subseteq \mathrm{cobal}T}$ which implies${\displaystyle \mathrm{cobal}(\mathrm{co}S)=\mathrm{cobal}S=\mathrm{cobal}(\mathrm{bal}S).}$

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

Given a fixed real number 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${\displaystyle r,s\ge 0}$ are non-negative scalars satisfying${\displaystyle r^p+s^p=1.}$ It is called anabsolutely${\displaystyle p}$-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${\displaystyle |r{|}^p+|s{|}^p\le 1.}$^[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:${\displaystyle q(x+y)\le q(x)+q(y)}$ 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 ofseminorms 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 notseminorms. For example, whenever then the map${\displaystyle q(f)={\int }_{\mathbb{R}}|f(t){|}^{p}dt}$ used to define theLp space${\displaystyle L_p(\mathbb{R})}$ is a${\displaystyle p}$-seminorm but not a seminorm.^[6]

Given atopological vector space is${\displaystyle p}$-seminormable (meaning that its topology is induced by some${\displaystyle p}$-seminorm) if and only if it has abounded${\displaystyle p}$-convex neighborhood of the origin.^[5]

The WikibookAlgebra 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.

• Abstract algebra
• Convex analysis
• Convex geometry
• Group theory
• Linear algebra

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