Minkowski Functional of Open Convex Set in Normed Vector Space is Sublinear Functional/Lemma

Lemma

Let $\struct {X, \norm \cdot}$ be a normed vector space.

Let $C$ be an open convex subset of $X$ with $0 \in C$.

Then if:

$\lambda \in \set {t > 0 : t^{-1} x \in C}$

we have:

$\hointr \lambda \infty \subseteq \set {t > 0 : t^{-1} x \in C}$

Further:

$\openint {\map {p_C} x} \infty \subseteq \set {t > 0 : t^{-1} x \in C}$

where $p_C$ is the Minkowski functional of $C$.

Proof

Let:

$t \in \set {t > 0 : t^{-1} x \in C}$

Then:

$t^{-1} x \in C$

Then from convexity, for all $\alpha \ge 1$ we have:

$\alpha^{-1} t^{-1} x + \paren {1 - \alpha^{-1} } \times 0 \in C$

that is:

$\alpha t \in \set {t > 0 : t^{-1} x \in C}$

so:

$\hointr t \infty \subseteq \set {t > 0 : t^{-1} x \in C}$

Now let:

$\lambda \in \openint {\map {p_C} x} \infty$

Then:

$\map {p_C} x < \lambda$

From the definition of infimum, there exists:

$\alpha \in \set {t > 0 : t^{-1} x \in C}$

such that:

$\map {p_C} x < \alpha < \lambda$

Then, from the above computation we have:

$\openint \alpha \infty \subseteq \set {t > 0 : t^{-1} x \in C}$

And, since $\alpha < \lambda$, we have:

$\lambda \in \openint \alpha \infty$

so:

$\lambda \in \set {t > 0 : t^{-1} x \in C}$

So, we obtain:

$\openint {\map {p_C} x} \infty \subseteq \set {t > 0 : t^{-1} x \in C}$

$\blacksquare$