absorbing set

Let V be a vector spaceMathworldPlanetmath over a field F equipped with a non-discrete valuationMathworldPlanetmath ||:F. 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 λF with |λ|r, BλA. 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 xV, there is a non-negative real number r such that xλA for all λF with |λ|r. If a finite subset B of V consists of x1,,xn, then corresponding to each xi, there is an ri0 such that xiλA such that |λri, λF. So xiλA with |λ|r if we take r=max{r1,,rn}. So A absorbs B.

Example. If V=n and F=, 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 λCC for all |λ|1. Clearly, C absorbs itself (Cλ-1C, |λ-1|1), and 0C. C is also symmetricPlanetmathPlanetmath (-C=C), for -CC and C=-(-C)-C. As an example of a circled set that is neither absorbing nor convex, consider V=2 and F=, 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