Definition:Minkowski Functional/Normed Vector Space
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Definition
Let $\Bbb F \in \set {\R, \C}$.
Let $\struct {X, \norm \cdot}$ be a normed vector space over $\Bbb F$.
Let $C$ be an open convex subset of $X$ with $0 \in C$.
The Minkowski functional of $C$ is the mapping $p_C : X \to \hointr 0 \infty$ defined as:
- $\forall x \in X: \map {p_C} x = \inf \set {t > 0 : \dfrac x t \in C}$
Also see
- Results about Minkowski functionals in normed vector spaces can be found here.
Source of Name
This entry was named for Hermann Minkowski.
Sources
- 2020: James C. Robinson: Introduction to Functional Analysis ... (previous) ... (next) $21.1$: The Minkowski Functional