# Definition:Minkowski Functional

Let $E$ be a vector space over $\R$.
A functional $p: E \to \R$ is called a Minkowski functional if it satisfies:
 $(1):\quad$ $\, \displaystyle \forall x \in E, \forall \lambda \in \R_{>0}: \,$ $\displaystyle p \left({\lambda x}\right)$ $=$ $\displaystyle \lambda p \left({x}\right)$ that is, $p$ is positive homogeneous $(2):\quad$ $\, \displaystyle \forall x, y \in E: \,$ $\displaystyle p \left({x + y}\right)$ $\le$ $\displaystyle p \left({x}\right) + p \left({y}\right)$ that is, $p$ is sub-additive