Let $X$ be a $\mathbb K$-vector space ($\mathbb K=\mathbb R$ or $\mathbb C$) and suppose $A\subset X$ is convex (and absorbing). How to show $$\{x\in X: \mu_A(x)<1\}\subset A?$$ Above $$\mu_A(x)=\inf\{\alpha>0: \alpha^{-1} x\in A\},$$ is the Minkowski functional of $A$.
What I tried was: Let $x\in X$ such that $\mu_A(x)<1$. By the infimum definition there exists $\alpha>0$ such that $\alpha^{-1} x\in A$ and $$\mu_A(x)\leq \alpha<1.$$ But I couldn't conclude the result from here. I don't know how the convexity will come into play.
If there is an $0 < \alpha < 1$ such that $\alpha^{-1}x \in A$, then
$$x = (1-\alpha)\cdot 0 + \alpha\cdot(\alpha^{-1}x) \in A,$$
since $A$ is convex, and $0\in A$ (because $A$ is absorbing).