Minkowski Functional of Open Convex Set 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 i  I f:


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

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:


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