In mathematics, an asymmetric norm on a vector space is a generalization of the concept of a norm.
Let X be a real vector space. Then an asymmetric norm on X is a function p : X → R satisfying the following properties:
non-negativity: for all x ∈ X, p(x) ≥ 0;definiteness: for x ∈ X, x = 0 if and only if p(x) = p(−x) = 0;homogeneity: for all x ∈ X and all λ ≥ 0, p(λx) = λp(x);the triangle inequality: for all x, y ∈ X, p(x + y) ≤ p(x) + p(y).On the real line R, the function p given byis an asymmetric norm but not a norm.
More generally, given a convex absorbing subset K of a real vector space containing no non-zero subspace, the Minkowski functional p given byis an asymmetric norm but not necessarily a norm, unless
K is also balanced.
