# Minkowski functional

Let $X$ be a normed space and let $K$ an absorbing convex subset of $X$ such that $0$ is in the interior of $K$. Then the Minkowski functional $\rho\colon X\to\mathbb{R}$ is defined as

 $\rho(x)=\inf\{\lambda>0\colon x\in\lambda K\}.$

We put $\rho(x)=0$ whenever $x=0$. Clearly $\rho(x)\geq 0$ for all $x$.

It is important to note that in general $\rho(x)\neq\rho(-x)$.

Properties
$\rho$
is positively $1$- homogeneous. This means that

 $\rho(s\cdot x)=s\cdot\rho(x)$

for $s>0$.

Title Minkowski functional MinkowskiFunctional 2013-03-22 14:50:44 2013-03-22 14:50:44 Mathprof (13753) Mathprof (13753) 18 Mathprof (13753) Definition msc 46B20