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 $$p : X \to [0, +\infty)$$ that has the following properties:


 * Subadditivity, or the triangle inequality: $$p(x + y) \leq p(x) + p(y) \text{ for all } x, y \in X.$$
 * Nonnegative homogeneity: $$p(rx) = r p(x) \text{ for all } x \in X$$ and every non-negative real number $$r \geq 0.$$
 * 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 $$x \neq 0,$$ 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 $$p(x) = \begin{cases}|x|, & x \leq 0; \\ 2 |x|, & x \geq 0; \end{cases}$$ is an asymmetric norm but not a norm.

In a real vector space $$X,$$ the $$p_B$$ of a convex subset $$B\subseteq X$$ that contains the origin is defined by the formula $$p_B(x) = \inf \left\{r \geq 0: x \in r B \right\}\,$$ for $$x \in X$$. This functional is an asymmetric seminorm if $$B$$ is an absorbing set, which means that $$\bigcup_{r \geq 0} r B = X,$$ 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 $$p(x) = \max_{\varphi \in B^*} \langle\varphi, x \rangle.$$ For instance, if $$B^* \subseteq \R^2$$ is the square with vertices $$(\pm 1,\pm 1),$$ then $$p$$ is the taxicab norm $$x = \left(x_0, x_1\right) \mapsto \left|x_0\right| + \left|x_1\right|.$$ 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 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 $$p(x) = \max_{\varphi \in B^*} \varphi(x)$$ is an asymmetric seminorm on $$X.$$