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 $X$ is a function $p:X\to [0,+\mathrm{\infty })$ that has the following properties:

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

Asymmetric norms differ from norms in that they need not satisfy the equality $p(-x)=p(x).$ 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 $x\ne 0,$ at least one of the two numbers $p(x)$ and $p(-x)$ is not zero.

Examples

On the real line $\mathbb{ℝ},$ the function $p$ given by $$p(x)=\{\begin{array}{ll}|x|,& x\le 0;\\ 2|x|,& x\ge 0;\end{array}$$ is an asymmetric norm but not a norm. In a real vector space $X,$ the Minkowski functional $p_B$ of a convex subset $B\subseteq X$ that contains the origin is defined by the formula $$p_B(x)=\inf \{r\ge 0:x\in rB\}$$ for $x\in X$. This functional is an asymmetric seminorm if $B$ is an absorbing set, which means that $\bigcup _{r\ge 0}rB=X,$ and ensures that $p(x)$ is finite for each $x\in X.$

Corresponce between asymmetric seminorms and convex subsets of the dual space

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

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

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

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.