Balanced set

In linear algebra and related areas of mathematics a balanced set, circled set or disk in a vector space (over a field formula_1 with an absolute value function formula_2) is a set formula_3 such that formula_4 for all scalars formula_5 satisfying formula_6 
The balanced hull or balanced envelope of a set formula_3 is the smallest balanced set containing formula_8
The balanced core of a subset formula_3 is the largest balanced set contained in formula_8 
Definition.
Suppose that formula_11 is a vector space over the field formula_1 of real or complex numbers. 
Elements of formula_1 are called scalars. 
Notation: If formula_3 is a set, formula_5 is a scalar, and formula_16 then let 
and for any formula_19 let 
denote the closed ball (respectively, the open ball) of radius formula_22 in formula_1 centered at formula_24 where formula_25 formula_26 and formula_27 
Every balanced subset of the field formula_1 is of the form formula_29 or formula_30 for some formula_31 
A subset formula_3 of formula_11 is called balanced if it satisfies any of the following equivalent conditions:
If formula_3 is a convex set then this list may be extended to include:
If formula_35 then this list may be extended to include:
The of a subset formula_3 of formula_37 denoted by formula_38 is defined in any of the following equivalent ways:
The of a subset formula_3 of formula_37 denoted by formula_41 is defined in any of the following equivalent ways: