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# 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 $s>0$.

Keywords:

Minkowski

Type of Math Object:

Definition

Major Section:

Reference

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