Definition:Minkowski Functional

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Let $E$ be a vector space over $\R$.

A functional $p: E \to \R$ is called a Minkowski functional if it satisfies:

\(\text {(1)}: \quad\) \(\, \displaystyle \forall x \in E, \forall \lambda \in \R_{>0}: \, \) \(\displaystyle p \left({\lambda x}\right)\) \(=\) \(\displaystyle \lambda p \left({x}\right)\) that is, $p$ is positive homogeneous
\(\text {(2)}: \quad\) \(\, \displaystyle \forall x, y \in E: \, \) \(\displaystyle p \left({x + y}\right)\) \(\le\) \(\displaystyle p \left({x}\right) + p \left({y}\right)\) that is, $p$ is sub-additive

Source of Name

This entry was named for Hermann Minkowski.