Minkowski Functional of Symmetric Convex Absorbing Set in Real Vector Space is Seminorm

Theorem

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

Let $A \subseteq X$ be a set that is symmetric, convex and absorbing.

Let $\mu_A$ be the Minkowski functional of $A$.

Then $\mu_A$ is a seminorm.

Proof

From Minkowski Functional of Convex Absorbing Set is Positive Homogeneous, $\mu_A$ is a sublinear functional.

Hence we have:

$\map {\mu_A} {x + y} \le \map {\mu_A} x + \map {\mu_A} y$ for all $x, y \in X$

and hence Seminorm Axiom $\text N 3$: Triangle Inequality.

We also have:

$\map {\mu_A} {r x} = r \map {\mu_A} x$ for all $r \ge 0$ and $x \in X$.

To establish Seminorm Axiom $\text N 2$: Positive Homogeneity, it remains to show that:

$\map {\mu_A} {r x} = \size r \map {\mu_A} x$ for all $r < 0$ and $x \in X$.

For this we first show that:

$\map {\mu_A} {-x} = \map {\mu_A} x$ for all $x \in X$.

Since $A$ is symmetric, we have $A = -A$.

Then we have:

\(\ds \map {\mu_A} x\)

\(=\)

\(\ds \inf \set {t > 0 : x \in t A}\)

Definition of Minkowski Functional of Convex Absorbing Set

\(\ds \)

\(=\)

\(\ds \inf \set {t > 0 : x \in -t A}\)

since $A = -A$

\(\ds \)

\(=\)

\(\ds \inf \set {t > 0 : -x \in t A}\)

\(\ds \)

\(=\)

\(\ds \map {\mu_A} {-x}\)

Now for $r < 0$ and $x \in X$ we have:

\(\ds \map {\mu_A} {r x}\)

\(=\)

\(\ds \map {\mu_A} {-\size r x}\)

\(\ds \)

\(=\)

\(\ds \size r \map {\mu_A} {-x}\)

Definition of Sublinear Functional

\(\ds \)

\(=\)

\(\ds \size r \map {\mu_A} x\)

showing Seminorm Axiom $\text N 2$: Positive Homogeneity.

$\blacksquare$