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