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| Let $V$ be a vector space over a field $F$ equipped with a |
Let $V$ be a vector space over a field $F$ equipped with a |
| non-discrete valuation $\abs{\cdot}:F\to\mathbb{R}$. Let $A,B$ be |
non-discrete valuation $\abs{\cdot}:F\to\mathbb{R}$. Let $A,B$ be |
| two subsets of $V$. Then $A$ is said to \emph{absorb} $B$ if there |
two subsets of $V$. Then $A$ is said to \emph{absorb} $B$ if there |
| is a non-negative real number $r$ such that, for all $\lambda\in F$ |
is a non-negative real number $r$ such that, for all $\lambda\in F$ |
| with $\abs{\lambda}\geq r$, $B\subseteq\lambda A$. $A$ is said to |
with $\abs{\lambda}\geq r$, $B\subseteq\lambda A$. $A$ is said to |
| be an \emph{absorbing set}, or a \emph{radial subset} of $V$ if $A$ |
be an \emph{absorbing set}, or a \emph{radial subset} of $V$ if $A$ |
| absorbs all finite subsets of $V$. |
absorbs all finite subsets of $V$. |
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| Equivalently, $A$ is absorbing if for any $x\in V$, there is a |
\textbf{Remarks}. |
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\begin{itemize} |
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\item 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 |
non-negative real number $r$ such that $x\in\lambda A$ for all |
| $\lambda\in F$ with $\abs{\lambda}\geq r$. If a finite subset $B$ |
$\lambda\in F$ with $\abs{\lambda}\geq r$. If a finite subset $B$ |
| of $V$ consists of $x_1,\ldots,x_n$, then corresponding to each |
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 |
$x_i$, there is an $r_i\geq 0$ such that $x_i\in\lambda A$ such that |
| $\vert\lambda\mid\geq r_i$, $\forall\lambda\in F$. So |
$\vert\lambda\mid\geq r_i$, $\forall\lambda\in F$. So |
| $x_i\in\lambda A$ with $\abs{\lambda}\geq r$ if we take |
$x_i\in\lambda A$ with $\abs{\lambda}\geq r$ if we take |
| $r=\max\lbrace r_1,\ldots,r_n\rbrace$. So $A$ absorbs $B$. |
$r=\max\lbrace r_1,\ldots,r_n\rbrace$. So $A$ absorbs $B$. |
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\item If $A$ is an absorbing set of $V$, then $A$ necessarily contains |
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$0$. For we know that $0\in\lambda A$. Since the valuation is |
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non-discrete, we can find a non-zero $\lambda$ large enough so that |
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$\abs{\lambda}\geq r$, so this implies that $0=\lambda^{-1}0\in |
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\lambda^{-1}\lambda A=A$. |
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\end{itemize} |
| \textbf{Example}. If $V=\mathbb{R}^n$ and $F=\mathbb{R}$, then any |
\textbf{Example}. If $V=\mathbb{R}^n$ and $F=\mathbb{R}$, then any |
| set containing an open ball centered at $0$ is absorbing. This |
set containing an open ball centered at $0$ is absorbing. This |
| implies that an absorbing set does not have to be connected, convex. |
implies that an absorbing set does not have to be connected, convex. |
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| A closely related concept is that of a \emph{circled set}, or a \emph{balanced set}. Let $V$ and $F$ be defined as above. A subset $C$ of $V$ is said to be |
A closely related concept is that of a \emph{circled set}, or a \emph{balanced set}. Let $V$ and $F$ be defined as above. A subset $C$ of $V$ is said to be |
| \emph{circled}, or \emph{balanced}, if $\lambda C\subseteq C$ for all $\abs{\lambda}\leq 1$. Clearly, $C$ absorbs itself ($C\subseteq\lambda^{-1}C$, |
\emph{circled}, or \emph{balanced}, if $\lambda C\subseteq C$ for all $\abs{\lambda}\leq 1$. Clearly, $C$ absorbs itself ($C\subseteq\lambda^{-1}C$, |
| $\abs{\lambda^{-1}}\geq 1$), and $0\in C$. $C$ is also symmetric |
$\abs{\lambda^{-1}}\geq 1$), and $0\in C$. $C$ is also symmetric |
| ($-C=C$), for $-C\subseteq C$ and $C=-(-C)\subseteq -C$. As an |
($-C=C$), for $-C\subseteq C$ and $C=-(-C)\subseteq -C$. As an |
| example of a circled set that is neither absorbing nor convex, |
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 |
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 |
$x$ 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 |
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. |
that is large enough so it is disjoint from the disk. |
