Definition:Minkowski Functional

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Definition

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

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

$(1):$	$\displaystyle $	$\displaystyle \forall x \in E, \forall \lambda \in \R, \lambda > 0:$	$\displaystyle $	$\displaystyle p(\lambda x)$	$=$	$\displaystyle \lambda p(x)$	$\displaystyle $	$\displaystyle $	$\displaystyle $	that is, $p$ is positive homogeneous
$(2):$	$\displaystyle $	$\displaystyle \forall x, y \in E:$	$\displaystyle $	$\displaystyle p(x+y)$	$\le$	$\displaystyle p(x) + p(y)$	$\displaystyle $	$\displaystyle $	$\displaystyle $	that is, $p$ is sub-additive

This entry was named for Hermann Minkowski.

\((1):\)	\(\displaystyle \)	\(\displaystyle \forall x \in E, \forall \lambda \in \R, \lambda > 0:\)	\(\displaystyle \)	\(\displaystyle p(\lambda x)\)	\(=\)	\(\displaystyle \lambda p(x)\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	that is, $p$ is positive homogeneous
\((2):\)	\(\displaystyle \)	\(\displaystyle \forall x, y \in E:\)	\(\displaystyle \)	\(\displaystyle p(x+y)\)	\(\le\)	\(\displaystyle p(x) + p(y)\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	that is, $p$ is sub-additive