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properties of Minkowski's functional
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(Theorem)
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Let $X$ be a normed space, $K$ convex subset of $X$ and $0$ belongs to the interior of $K$ .Then
- $\rho_{K}(x)\geq 0$ for all $x\in X$
- $\rho_{K}(0)= 0$
- $\rho_{K}(\lambda x)= \lambda \rho_{K}(x)$ , for all $\lambda\geq 0$ and $x\in X$
- $\rho_{K}(x+y)\leq \rho_{K}(x)+\rho_{K}(y)$ for all $x,y \in K$
- $\{x\in X\colon \rho_{K}(x)<1\}\subset K \subset \{x\in X\colon \rho_{K}(x)\leq 1\}$
- $K^{0}=\{x\in X\colon \rho_{K}(x)<1\}$ where $K^{0}$ denotes the interior of $K$
- $\bar K=\{x\in X\colon \rho_{K}(x)\leq 1\}$ where $\bar K$ denotes the closure of $K$
- $Bd(K)= \{x\in X\colon \rho_{K}(x)= 1\}$ where the $Bd(K)$ denotes the boundary of $K$ .
Minkowski's functional is a useful tool to prove propositions and solve exercises. Let us see an example
Example Let $K$ be a convex subset of $X$ . Show that $Ex(K)\subset Bd(K)$ , where $Ex(K)$ denotes the set of extreme points of $K$ .
If $x\in Ex(K)$ then from this follows that $x\in 1K$ and $\rho_{K}(x)= 1$ . Now we hypothesize that $\rho_{K}(x)<1$ then there is a real number $s$ such that $\rho_{K}(x)<s<1$ and so $\rho_{K}(\frac{x}{s})<1$ . Therefore we have that $x=s\frac{x}{s}+(1-s)0 \in K$ , that contradicts to the fact that $x\in Ex(K).$
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"properties of Minkowski's functional" is owned by georgiosl.
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Cross-references: real number, extreme points, propositions, Minkowski's functional, boundary, closure, interior, convex subset, normed space
This is version 7 of properties of Minkowski's functional, born on 2006-03-09, modified 2006-10-09.
Object id is 7704, canonical name is PropertiesOfMinkowskisFunctional.
Accessed 1215 times total.
Classification:
| AMS MSC: | 46B20 (Functional analysis :: Normed linear spaces and Banach spaces; Banach lattices :: Geometry and structure of normed linear spaces) |
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Pending Errata and Addenda
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