# Minkowski functional of a convex set is a convex function.

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!

• Perhaps you need the further property that $x\in K\Rightarrow -x\in K$? Nov 17 '15 at 4:23
• @charlestoncrabb We don't have this property in this question... Nov 17 '15 at 4:25