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Let $X$ be a real vector space, and $K$ be a convex set with $2$ properties: $0\in K$ and $\forall x\in X, \exists t >0$ s.t. $x/t\in K$.

Define the Minkowski functional of the set $K$ to be $p_K(x)=\inf \{t>0: x/t\in K \}$. Show that $p_K(x)$ is convex.

I have tried for some while using the definition of a convex function, but failed to prove this fact. Any idea on how to show it?

Thanks!

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  • $\begingroup$ Perhaps you need the further property that $x\in K\Rightarrow -x\in K$? $\endgroup$ Nov 17 '15 at 4:23
  • $\begingroup$ @charlestoncrabb We don't have this property in this question... $\endgroup$
    – Toad Jiang
    Nov 17 '15 at 4:25
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Hint: Try to show Triangle Inequality first.

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