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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.
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 spacePages displaying wikidata descriptions as a fallback