Doubt about Minkowski's Functional

Asked Sep 25 '18 at 22:49

Active Sep 25 '18 at 23:27

Viewed 135 times

Definition: Let $C$ be an open set in a real normed space $X$ with $0\in C.$ The Minkowski functional of $C$ is defined by $$p_{C}(x) = \inf\{\alpha>0:\alpha^{-1}x\in C\},x\in X.$$ Now I am trying to show the following

Lemma: Let $C$ and $X$ be as defined earlier and in addition suppose that $C$ is convex, then $p_C$ is a sublinear fuctional on X with $$C=\{x:p_C(x)<1\}.$$ In the proof the author starts of by saying, for $x,y\in X$ choose arbritary $\alpha>p_C(x)$ and $\beta>p_C(y).$ Then $\alpha^{-1}x\in C$ and $\beta^{-1}y\in C.$ I don't understand how this can be deduced from the definition of $p_C.$ Perhaps someone can explain?

edited Sep 25 '18 at 23:06

asked Sep 25 '18 at 22:49

Student

9,366

1

I gave an answer showing that $\alpha >p_C(x)$ implies $(\alpha)^{-1}x\in C$. All you need for this is definition of infimum of a set, convexity of $C$ and the fact that $0\in C$. Can someone say why I was voted down without an explanation? – Kavi Rama Murthy Sep 26 '18 at 05:46
3

Does this answer your question? Norm induced by convex, open, symmetric, bounded set in $\Bbb R^n$. – ViktorStein Jun 06 '20 at 14:34

Doubt about Minkowski's Functional

0 Answers0