Definition:Minkowski Functional

Definition

Let $\GF \in \set {\R, \C}$.

Let $X$ be a vector space over $\GF$.

Let $A \subseteq X$ be a convex absorbing subset of $X$.

The Minkowski functional of $A$ is the real-valued function $\mu_A : X \to \closedint 0 \infty$ defined as:

$\forall x \in X: \map {\mu_A} x = \inf \set {t > 0 : \dfrac x t \in A}$

Normed Vector Space

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 Convex Absorbing Set is Finite

• Results about Minkowski functionals can be found here.

Source of Name

This entry was named for Hermann Minkowski.

Sources

• 1991: Walter Rudin: Functional Analysis (2nd ed.) ... (previous) ... (next): $1.33$: Definitions