Asymmetric norm

Examples

On the real line ${\displaystyle \mathbb{R},}$ the function ${\displaystyle p}$ given by $${\displaystyle p(x)=\{\begin{array}{ll}|x|,& x\le 0;\\ 2|x|,& x\ge 0;\end{array}}$$ 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 $${\displaystyle p_B(x)=inf\left\{r\ge 0:x\in rB\right\}}$$ for ${\displaystyle x\in X}$. This functional is an asymmetric seminorm if ${\displaystyle B}$ is an absorbing set, which means that ${\displaystyle \bigcup _{r\ge 0}rB=X,}$ and ensures that ${\displaystyle p(x)}$ is finite for each ${\displaystyle x\in X.}$

Corresponce between asymmetric seminorms and convex subsets of the dual space

If ${\displaystyle B^{\ast }\subseteq {\mathbb{R}}^n}$ is a convex set that contains the origin, then an asymmetric seminorm ${\displaystyle p}$ can be defined on ${\displaystyle {\mathbb{R}}^n}$ by the formula $${\displaystyle p(x)=\underset{\phi \in B^{\ast }}{max}⟨\phi ,x⟩.}$$ For instance, if ${\displaystyle B^{\ast }\subseteq {\mathbb{R}}^2}$ is the square with vertices ${\displaystyle (\pm 1,\pm 1),}$ then ${\displaystyle p}$ is the taxicab norm ${\displaystyle x=\left(x_0,x_1\right)\mapsto \left|x_0\right|+\left|x_1\right|.}$ Different convex sets yield different seminorms, and every asymmetric seminorm on ${\displaystyle {\mathbb{R}}^n}$ 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 ${\displaystyle p}$ is

• positive definite if and only if ${\displaystyle B^{\ast }}$ contains the origin in its topological interior,
• degenerate if and only if ${\displaystyle B^{\ast }}$ is contained in a linear subspace of dimension less than ${\displaystyle n,}$ and
• symmetric if and only if ${\displaystyle B^{\ast }=-B^{\ast }.}$

More generally, if ${\displaystyle X}$ is a finite-dimensional real vector space and ${\displaystyle B^{\ast }\subseteq X^{\ast }}$ is a compact convex subset of the dual space ${\displaystyle X^{\ast }}$ that contains the origin, then ${\displaystyle p(x)=\underset{\phi \in B^{\ast }}{max}\phi (x)}$ is an asymmetric seminorm on ${\displaystyle X.}$