Editing Seminorm (section)

==Minkowski functionals and seminorms==
{{Main|Minkowski functional}}

Seminorms on a vector space <math>X</math> are intimately tied, via Minkowski functionals, to subsets of <math>X</math> that are [[Convex set|convex]], [[Balanced set|balanced]], and [[Absorbing set|absorbing]].  Given such a subset <math>D</math> of <math>X,</math> the Minkowski functional of <math>D</math> is a seminorm.  Conversely, given a seminorm <math>p</math> on <math>X,</math> the sets<math>\{x \in X : p(x) < 1\}</math> and <math>\{x \in X : p(x) \leq 1\}</math> are convex, balanced, and absorbing and furthermore, the Minkowski functional of these two sets (as well as of any set lying "in between them") is <math>p.</math>{{sfn|Schaefer|Wolff|1999|p=40}}