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 is a function that has the following properties:

• Subadditivity, or the triangle inequality:
• Nonnegative homogeneity: and every non-negative real number
• Positive definiteness:

Asymmetric norms differ from norms in that they need not satisfy the equality

If the condition of positive definiteness is omitted, then is an asymmetric seminorm. A weaker condition than positive definiteness is non-degeneracy: that for at least one of the two numbers and is not zero.

Examples

On the real line the function given by

is an asymmetric norm but not a norm.

In a real vector space the Minkowski functional of a convex subset that contains the origin is defined by the formula

for This functional is an asymmetric seminorm if is an absorbing set, which means that and ensures that is finite for each

Corresponce between asymmetric seminorms and convex subsets of the dual space

If is a convex set that contains the origin, then an asymmetric seminorm can be defined on by the formula

For instance, if is the square with vertices then is the taxicab norm Different convex sets yield different seminorms, and every asymmetric seminorm on 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 is

• positive definite if and only if contains the origin in its topological interior,
• degenerate if and only if is contained in a linear subspace of dimension less than and
• symmetric if and only if

More generally, if is a finite-dimensional real vector space and is a compact convex subset of the dual space that contains the origin, then is an asymmetric seminorm on

See also

• Finsler manifold – smooth manifold equipped with a Minkowski functional at each tangent space
• Minkowski functional – Function made from a set