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

Definition

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).

Examples

• On the real line R, the function p given by

p(x) = \begin{cases} |x|, & x \leq 0; \\ 2 |x|, & x \geq 0; \end{cases}

is an asymmetric norm but not a norm.

• More generally, given a strictly positive function g : Sⁿ⁻¹   R defined on the unit sphere Sⁿ⁻¹ in Rⁿ (with respect to the usual Euclidean norm | |, say), the function p given by

p(x) = g(x/|x|) |x| \,

is an asymmetric norm on Rⁿ but not necessarily a norm.

References