# 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 $|\lambda |\ge r$, $B\subseteq \lambda A$. $A$ 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 $|\lambda |\ge r$. If a finite subset $B$ of $V$ consists of ${x}_{1},\mathrm{\dots},{x}_{n}$, then corresponding to each ${x}_{i}$, there is an ${r}_{i}\ge 0$ such that ${x}_{i}\in \lambda A$ such that $|\lambda \mid \ge {r}_{i}$, $\forall \lambda \in F$. So ${x}_{i}\in \lambda A$ with $|\lambda |\ge r$ if we take $r=\mathrm{max}\{{r}_{1},\mathrm{\dots},{r}_{n}\}$. 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 $|\lambda |\le 1$. Clearly, $C$ absorbs itself ($C\subseteq {\lambda}^{-1}C$,
$|{\lambda}^{-1}|\ge 1$), 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 |