Definition:Minkowski Functional

Definition

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

This entry was named for Hermann Minkowski.

\((1)\)	$:$			\(\ds \forall x \in E, \forall \lambda \in \R_{>0}:\)		\(\ds \map p {\lambda x} \)	\(\ds = \)	\(\ds \lambda \map p x \)	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