Minkowski Functional of Open Convex Set is Sublinear Functional

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

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

Let $p_C$ be the Minkowski functional for $C$.

Then $p_C$ is a sublinear functional.

Proof
We will show that:


 * $(1): \quad$ $\map {p_C} {\lambda x} = \lambda \map {p_C} x$ for each $x \in X$ and $\lambda \in \R_{\ge 0}$
 * $(2): \quad$ $\map {p_C} {x + y} \le \map {p_C} x + \map {p_C} y$ for each $x, y \in X$.

Proof of $(1)$
If $\lambda = 0$, then $(1)$ follows immediately since:


 * $\map {p_C} 0 = 0$

as shown in Minkowski Functional of Open Convex Set is Well-Defined.

Now take $\lambda \ne 0$.

We then have, for each $x \in X$:


 * $t^{-1} x \in C$ $\paren {\lambda t}^{-1} \paren {\lambda x} \in C$

So:


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

giving:


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

So, from Multiple of Infimum, we have:


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

Then from the definition of the Minkowski functional, we have:


 * $\map {p_C} {\lambda x} = \lambda \map {p_C} x$

Lemma
Now let $x, y \in X$.

We show that:


 * $\map {p_C} {x + y} \le \map {p_C} x + \map {p_C} y$

Let $\epsilon > 0$.

Pick $\alpha > 0$ such that:


 * $\ds \map {p_C} x < \alpha < \map {p_C} x + \frac \epsilon 2$

and pick $\beta$ such that:


 * $\ds \map {p_C} y < \beta < \map {p_C} y + \frac \epsilon 2$

Then we have:


 * $\alpha^{-1} x \in C$

and:


 * $\beta^{-1} y \in C$

Note that we have:


 * $\ds \frac \alpha {\alpha + \beta} + \frac \beta {\alpha + \beta} = 1$

So, from convexity, we have:


 * $\ds \frac \alpha {\alpha + \beta} \paren {\alpha^{-1} x} + \frac \beta {\alpha + \beta} \paren {\beta^{-1} y} \in C$

That is:


 * $\ds \frac {x + y} {\alpha + \beta} \in C$

So:


 * $\alpha + \beta \in \set {t > 0 : t^{-1} \paren {x + y} \in C}$

From the definition of infimum, we have:


 * $\map {p_C} {x + y} \le \alpha + \beta < \map {p_C} x + \map {p_C} y + \epsilon$

Since $\epsilon > 0$ was arbitrary, we have:


 * $\map {p_C} {x + y} \le \map {p_C} x + \map {p_C} y$

Since $(1)$ and $(2)$ hold, we have that:


 * $p_C$ is a sublinear functional.