Convex Absorbing Set contained between Sets in terms of Minkowski Functional

Theorem

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

Let $\struct {X, \tau}$ be a topological vector space over $\GF$.

Let $C$ be a convex absorbing set.

Let $\mu_C$ be the Minkowski functional of $C$.

Then we have:

$\set {x \in X : \map {\mu_C} x < 1} \subseteq C \subseteq \set {x \in X : \map {\mu_C} x \le 1}$

Proof

Let $x \in X$ be such that $\map {\mu_C} x < 1$.

Then:

$\inf \set {t > 0 : \dfrac x t \in C} < 1$

So there exists $t < 1$ such that $x \in t C$.

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

So, we have, since $C$ is convex:

$x = x + \paren {1 - t} {\mathbf 0}_X \in t C + \paren {1 - t} C \subseteq C$

So $x \in C$.

So we have:

$\set {x \in X : \map {\mu_C} x < 1} \subseteq C$

Now let $x \in C$.

We clearly have:

$1 \in \set {t > 0 : \dfrac x t \in C}$

So:

$\map {\mu_C} x = \inf \set {t > 0 : \dfrac x t \in C} \le 1$

Hence we obtain the inclusion:

$C \subseteq \set {x \in X : \map {\mu_C} x \le 1}$

$\blacksquare$