Asymmetric norm

An asymmetric norm on a real vector space ${\displaystyle X}$ is a function ${\displaystyle p:X\to [0,+\infty )}$ that has the following properties:

• Subadditivity, or the triangle inequality: ${\displaystyle p(x+y)\leq p(x)+p(y){\text{ for all }}x,y\in X.}$
• Nonnegative homogeneity: ${\displaystyle p(rx)=rp(x){\text{ for all }}x\in X}$ and every non-negative real number ${\displaystyle r\geq 0.}$
• Positive definiteness: ${\displaystyle p(x)>0{\text{ unless }}x=0}$

Asymmetric norms differ from norms in that they need not satisfy the equality ${\displaystyle p(-x)=p(x).}$

If the condition of positive definiteness is omitted, then ${\displaystyle p}$ is an asymmetric seminorm. A weaker condition than positive definiteness is non-degeneracy: that for ${\displaystyle x\neq 0,}$ at least one of the two numbers ${\displaystyle p(x)}$ and ${\displaystyle p(-x)}$ is not zero.

On the real line ${\displaystyle \mathbb {R} ,}$ the function ${\displaystyle p}$ given by $${\displaystyle p(x)={\begin{cases}|x|,&x\leq 0;\\2|x|,&x\geq 0;\end{cases}}}$$ is an asymmetric norm but not a norm.

In a real vector space ${\displaystyle X,}$ the Minkowski functional ${\displaystyle p_{B}}$ of a convex subset ${\displaystyle B\subseteq X}$ that contains the origin is defined by the formula $${\displaystyle p_{B}(x)=\inf \left\{r\geq 0:x\in rB\right\}\,}$$ for ${\displaystyle x\in X}$. This functional is an asymmetric seminorm if ${\displaystyle B}$ is an absorbing set, which means that ${\displaystyle \bigcup _{r\geq 0}rB=X,}$ and ensures that ${\displaystyle p(x)}$ is finite for each ${\displaystyle x\in X.}$