Definition:Minkowski Functional/Normed Vector Space

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

• Minkowski Functional of Open Convex Set in Normed Vector Space is Well-Defined

• 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