Minkowski Functional of Convex Absorbing Set is Positive Homogeneous

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

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

Let $A$ be a convex absorbing set.

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

Then $\mu_A$ is positive homogeneous.

That is, for each $t \ge 0$ we have:


 * $\map {\mu_A} {t x} = t \map {\mu_A} x$

Proof
From Absorbing Set in Vector Space contains Zero Vector, we have ${\mathbf 0}_X \in A$.

So $\map {\mu_A} {\mathbf 0_X} = 0$.

So the claim clearly holds for $t = 0$.

Now take $t > 0$.

We argue that:


 * $\set {s > 0 : t x \in s C} = t \set {s > 0 : x \in s C}$

We have:


 * $s \in \set {s > 0 : t x \in s C}$

$s > 0$ and:


 * $t x \in s C$

This is equivalent to:


 * $\ds x \in \paren {\frac s t} C$

which is equivalent to:


 * $\ds \frac s t \in \set {s > 0 : x \in s C}$

Hence we deduce:


 * $\set {s > 0 : t x \in s C} = t \set {s > 0 : x \in s C}$

From Multiple of Infimum, we obtain:


 * $\map {\mu_C} {t x} = t \map {\mu_C} x$