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 , +\infty) that has the following properties:

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

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 $x \neq 0,$ at least one of the two numbers $p(x)$ and $p(-x)$ is not zero.

Examples

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

In a real vector space $X,$ the $p_B$ of a convex subset $B\subseteq X$ that contains the origin is defined by the formula
$p_B(x) = \inf \left\\,$ for $x \in X$
This functional is an asymmetric seminorm if $B$ is an absorbing set, which means that $\bigcup_ r B = X,$ and ensures that $p(x)$ is finite for each $x \in X.$

Corresponce between asymmetric seminorms and convex subsets of the dual space

If $B^* \subseteq \R^n$ is a convex set that contains the origin, then an asymmetric seminorm $p$ can be defined on $\R^n$ by the formula
$p(x) = \max_ \langle\varphi, x \rangle.$
For instance, if $B^* \subseteq \R^2$ is the square with vertices $(\pm 1,\pm 1),$ then $p$ is the taxicab norm $x = \left(x_0, x_1\right) \mapsto \left, x_0\ + \left, x_1\.$ Different convex sets yield different seminorms, and every asymmetric seminorm on $\R^n$ 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 Interior (mathematics), 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 $p(x) = \max_ \varphi(x)$ is an asymmetric seminorm on $X.$

See also

*
*

References

*
* S. Cobzas, ''Functional Analysis in Asymmetric Normed Spaces'', Frontiers in Mathematics, Basel: Birkhäuser, 2013; .

Linear algebra
Norms (mathematics)

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