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 and only if it satisfies:

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

Source of Name

This entry was named for Hermann Minkowski.