# absorbing set

Let $V$ be a vector space over a field $F$ equipped with a non-discrete valuation $|{\cdot}|:F\to\mathbb{R}$. Let $A,B$ be two subsets of $V$. Then $A$ is said to absorb $B$ if there is a non-negative real number $r$ such that, for all $\lambda\in F$ with $|{\lambda}|\geq r$, $B\subseteq\lambda A$. $A$ is said to be an absorbing set, or a radial subset of $V$ if $A$ absorbs all finite subsets of $V$.

Equivalently, $A$ is absorbing if for any $x\in V$, there is a non-negative real number $r$ such that $x\in\lambda A$ for all $\lambda\in F$ with $|{\lambda}|\geq r$. If a finite subset $B$ of $V$ consists of $x_{1},\ldots,x_{n}$, then corresponding to each $x_{i}$, there is an $r_{i}\geq 0$ such that $x_{i}\in\lambda A$ such that $|\lambda\mid\geq r_{i}$, $\forall\lambda\in F$. So $x_{i}\in\lambda A$ with $|{\lambda}|\geq r$ if we take $r=\max\{r_{1},\ldots,r_{n}\}$. So $A$ absorbs $B$.

Example. If $V=\mathbb{R}^{n}$ and $F=\mathbb{R}$, then any set containing an open ball centered at $0$ is absorbing. This implies that an absorbing set does not have to be connected, convex.

A closely related concept is that of a circled set, or a balanced set. Let $V$ and $F$ be defined as above. A subset $C$ of $V$ is said to be circled, or balanced, if $\lambda C\subseteq C$ for all $|{\lambda}|\leq 1$. Clearly, $C$ absorbs itself ($C\subseteq\lambda^{-1}C$, $|{\lambda^{-1}}|\geq 1$), and $0\in C$. $C$ is also symmetric ($-C=C$), for $-C\subseteq C$ and $C=-(-C)\subseteq-C$. As an example of a circled set that is neither absorbing nor convex, consider $V=\mathbb{R}^{2}$ and $F=\mathbb{R}$, and $C$ the union of $x$ and $y$ axes. For an example of an absorbing set that is not circled, take the union of a unit disk and an annulus centered at 0 that is large enough so it is disjoint from the disk.

 Title absorbing set Canonical name AbsorbingSet Date of creation 2013-03-22 15:26:24 Last modified on 2013-03-22 15:26:24 Owner CWoo (3771) Last modified by CWoo (3771) Numerical id 10 Author CWoo (3771) Entry type Definition Classification msc 46A08 Classification msc 15A03 Related topic BalancedSet Related topic AbsorbingElement Defines absorbing Defines absorb Defines radial