Asymmetric norm

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 ${\displaystyle X}$ 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.

Examples

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 ${\displaystyle x\in X.}$

Corresponce between asymmetric seminorms and convex subsets of the dual space

If ${\displaystyle B^{*}\subseteq \mathbb {R} ^{n}}$ 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 $${\displaystyle p(x)=\max _{\varphi \in B^{*}}\langle \varphi ,x\rangle .}$$ For instance, if ${\displaystyle B^{*}\subseteq \mathbb {R} ^{2}}$ 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 ${\displaystyle B^{*}}$ contains the origin in its topological interior,
• degenerate if and only if ${\displaystyle B^{*}}$ is contained in a linear subspace of dimension less than ${\displaystyle n,}$ and
• symmetric if and only if ${\displaystyle B^{*}=-B^{*}.}$

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

See also

• Finsler manifold – Generalization of Riemannian manifolds
• Minkowski functional – Function made from a set

References

• Cobzaş, S. (2006). "Compact operators on spaces with asymmetric norm". Stud. Univ. Babeş-Bolyai Math. 51 (4): 69–87. arXiv:math/0608031. Bibcode:2006math......8031C. ISSN 0252-1938. MR 2314639.
• S. Cobzas, Functional Analysis in Asymmetric Normed Spaces, Frontiers in Mathematics, Basel: Birkhäuser, 2013; ISBN 978-3-0348-0477-6.