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 "V" is a function formula_1 that has the following properties:
Asymmetric norms differ from norms in that they need not satisfy the equality "p"(-"v") = "p"("v").
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 "v" ≠ 0, at least one of the two numbers "p"("v") and "p"(-"v") is not zero.
Corresponce between asymmetric seminorms and convex subsets of the dual space.
If formula_12 is a convex set that contains the origin, then an asymmetric seminorm formula_13 can be defined on formula_14 by the formula
For instance, if formula_16 is the square with vertices formula_17, then formula_13 is the taxicab norm formula_19. Different convex sets yield different seminorms, and every asymmetric seminorm on formula_14 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 formula_13 is 
More generally, if formula_3 is a finite-dimensional real vector space and formula_27 is a compact convex subset of the dual space formula_28 that contains the origin, then formula_29 is an asymmetric seminorm on formula_3.