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

Theorem

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

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 in Normed Vector Space is Well-Defined.

Now take $\lambda \ne 0$.

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

$t^{-1} x \in C$ if and only if $\paren {\lambda t}^{-1} \paren {\lambda x} \in C$

So:

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

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:

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

$\Box$

Proof of $(2)$

Lemma

If:

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

we have:

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

$\Box$

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:

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

$\Box$

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

$p_C$ is a sublinear functional.

$\blacksquare$

Sources

• 2020: James C. Robinson: Introduction to Functional Analysis ... (previous) ... (next) $21.1$: The Minkowski Functional