Let be a vector space over a field equipped with a non-discrete valuation . Let be two subsets of . Then is said to absorb if there is a non-negative real number such that, for all with , . is said to be an absorbing set, or a radial subset of if absorbs all finite subsets of .
Equivalently, is absorbing if for any , there is a non-negative real number such that for all with . If a finite subset of consists of , then corresponding to each , there is an such that such that , . So with if we take . So absorbs .
Example. If and , then any set containing an open ball centered at 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 and be defined as above. A subset of is said to be circled, or balanced, if for all . Clearly, absorbs itself (, ), and . is also symmetric (), for and . As an example of a circled set that is neither absorbing nor convex, consider and , and the union of and 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.
|Date of creation||2013-03-22 15:26:24|
|Last modified on||2013-03-22 15:26:24|
|Last modified by||CWoo (3771)|