In mathematics, an asymmetric norm on a vector space is a generalization of the concept of a norm.
An asymmetric norm on a real vector space
that has the following properties: 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
On the real line
{\displaystyle 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
the Minkowski functional
of a convex subset
that contains the origin is defined by the formula
This functional is an asymmetric seminorm if
is an absorbing set, which means that
is a convex set that contains the origin, then an asymmetric seminorm
max
is the square with vertices
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.
is a finite-dimensional real vector space and
is a compact convex subset of the dual space
that contains the origin, then
max
is an asymmetric seminorm on
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