In mathematics, an asymmetric norm on a vector space is a generalization of the concept of a norm.
An asymmetric norm on a real vector space is a function that has the following properties:
Asymmetric norms differ from norms in that they need not satisfy the equality
If the condition of positive definiteness is omitted, then is an asymmetric seminorm. A weaker condition than positive definiteness is non-degeneracy: that for at least one of the two numbers and is not zero.
On the real line the function given by
In a real vector space the Minkowski functional of a convex subset that contains the origin is defined by the formula
If is a convex set that contains the origin, then an asymmetric seminorm can be defined on by the formula
More generally, if is a finite-dimensional real vector space and is a compact convex subset of the dual space that contains the origin, then is an asymmetric seminorm on