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:

$s \in \set {s > 0 : t x \in s C}$

if and only if $s > 0$ and:

$t x \in s C$

This is equivalent to:

$x \in \paren {\dfrac s t} C$

which is equivalent to:

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

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:

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

$\blacksquare$