In mathematics, an asymmetric norm on a vector space is a generalization of the concept of a norm.

Definition

An asymmetric norm on a real vector space X 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 p is an asymmetric seminorm. A weaker condition than positive definiteness is non-degeneracy: that for x \neq 0, at least one of the two numbers p(x) and p(-x) is not zero.

Examples

On the real line \R, the function p given by is an asymmetric norm but not a norm. In a real vector space X, the p_B of a convex subset that contains the origin is defined by the formula for x \in X. This functional is an asymmetric seminorm if B is an absorbing set, which means that and ensures that p(x) is finite for each x \in X.

Corresponce between asymmetric seminorms and convex subsets of the dual space

If is a convex set that contains the origin, then an asymmetric seminorm p can be defined on \R^n by the formula For instance, if is the square with vertices then p is the taxicab norm Different convex sets yield different seminorms, and every asymmetric seminorm on \R^n can be obtained from some convex set, called its dual unit ball. Therefore, asymmetric seminorms are in one-to-one correspondence with convex sets that contain the origin. The seminorm p is More generally, if X is a finite-dimensional real vector space and is a compact convex subset of the dual space X^* that contains the origin, then is an asymmetric seminorm on X.