# 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*
$\rho :X\to \mathbb{R}$ is defined as

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

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

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

Properties

$\rho $ is positively $1$- homogeneous^{}. This means that

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

for $s>0$.

Title | Minkowski functional |
---|---|

Canonical name | MinkowskiFunctional |

Date of creation | 2013-03-22 14:50:44 |

Last modified on | 2013-03-22 14:50:44 |

Owner | Mathprof (13753) |

Last modified by | Mathprof (13753) |

Numerical id | 18 |

Author | Mathprof (13753) |

Entry type | Definition |

Classification | msc 46B20 |