Absolutely convex set

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.

If $S$ is a subset of a real or complex vector space $X,$ then we call $S$ a disk and say that $S$ is disked, absolutely convex, and convex balanced if any of the following equivalent conditions is satisfied:

1. $S$ is convex and balanced;
2. for any scalars $a$ and $b,$ if $|a|+|b|\leq 1$ then $aS+bS\subseteq S$;
3. for all scalars $a,b,$ and $c,$ if $|a|+|b|\leq |c|,$ then $aS+bS\subseteq cS$;
4. for any scalars $a_{1},\ldots ,a_{n}$ if $\sum _{i=1}^{n}\left|a_{i}\right|\leq 1$ then $a_{1}S+\cdots +a_{n}S\subseteq S$;
5. for any scalars $c,a_{1},\ldots ,a_{n}$ if $\sum _{i=1}^{n}\left|a_{i}\right|\leq |c|$ then $a_{1}S+\cdots +a_{n}S\subseteq cS$;

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

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

1. $\operatorname {co} \left(\operatorname {bal} S\right),$ which is the convex hull of the balanced hull of $S$; thus, $\operatorname {cobal} S=\operatorname {co} \left(\operatorname {bal} S\right)$;
   • Note however that in general, $\operatorname {cobal} S\neq \operatorname {bal} \left(\operatorname {co} S\right),$ even in finite dimensions,
2. the intersection of all disks containing $S,$
3. $\left\{\sum _{i=1}^{n}\lambda _{i}x_{i}~:~n\in \mathbb {N} ,\,x_{i}\in A,\,\sum _{i=1}^{n}\left|\lambda _{i}\right|\leq 1\right\},$ where the $\lambda _{i}$ are elements of the underlying field.

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 $X$ is absorbing in $X$ if and only if $\operatorname {span} D=X.$

Properties

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 $E+E\subseteq S.$
• The convex balanced hull of $S$ contains both the convex hull of $S$ and the balanced hull of $S.$
• The absolutely convex hull of a bounded set in a topological vector space is again bounded.
• If $D$ is a bounded disk in a TVS $X$ and if $x_{\bullet }=\left(x_{i}\right)_{i=1}^{\infty }$ is a sequence in $D,$ then the partial sums $s_{\bullet }=\left(s_{n}\right)_{n=1}^{\infty }$ are Cauchy, where for all $n,$ $s_{n}:=\sum _{i=1}^{n}2^{-i}x_{i}.$
  • 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 $D.$

Examples

Although $\operatorname {cobal} S=\operatorname {co} \left(\operatorname {bal} S\right),$ the convex balanced hull of $S$ is not necessarily equal to the balanced hull of the convex hull of $S.$ For an example where $\operatorname {cobal} S\neq \operatorname {bal} \left(\operatorname {co} S\right)$ let $X$ be the real vector space $\mathbb {R} ^{2}$ and let $S:=\{(-1,1),(1,1)\}.$ Then $\operatorname {bal} \left(\operatorname {co} S\right)$ is a strict subset of $\operatorname {cobal} S$ that is not even convex. In particular, this example also shows that the balanced hull of a convex set is not necessarily convex. To see this, note that $\operatorname {cobal} S$ is equal to the closed square in $X$ with vertices $(-1,1),(1,1),(-1,-1),$ and $(1,-1)$ while $\operatorname {bal} \left(\operatorname {co} S\right)$ is a closed "hour glass shaped" shaped subset that intersects the $x$-axis at the origin and is the union of two triangles: one whose vertices are the origin along with $S$ and the other triangle whose vertices are the origin along with $-S=\{(-1,-1),(1,-1)\}.$

See also

• Absorbing set – Set that can be "inflated" to reach any point
• 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

Bibliography

• 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, H.H. (1999). Topological vector spaces. Springer-Verlag Press. p. 39.
• Trèves, François (2006) [1967]. Topological Vector Spaces, Distributions and Kernels. Mineola, N.Y.: Dover Publications. ISBN 978-0-486-45352-1. OCLC 853623322.