Minkowski functional characterization for convex and absorbing sets Let $(X,\|\;\|)$ be a normed vector space over $K$. Let $E\subset X$ be convex and absorbing. And let $E_1=\{x\in X:p_E(x)<1\}$, $E_2=\{x\in X: p_E(x)\le 1\}$; where $p_E(x)=\inf_{x\in tE}\{t>0\}$. I want to prove that:
$$
E^\circ\subset E_1\subset E\subset E_2\subset\overline E
$$
I'm stucked figuring out how does every $x\in E^\circ$ should satisfy $p_E(x)<1$. I only got that for every $x\in E^\circ$ I can found $r_x>0$ s.t. $B_{r_x}(x)\subset E^\circ$, but doesn't seem to help. Any ideas would be appreciated.

I think I've managed to prove this first part ($E^\circ\subset E_1$):
Let $x\in E^\circ$, so there $\exists\;r_0>0$ s.t. $B_{r_0}(x)\subset E^\circ$. Taking $\tilde r_0=\min\{r_0,1\}$ we get that:
$$
\|(1+\tilde r_0)x-x\|=|\tilde r_0|\|x\|\le\|x\|<r_0\;\Rightarrow (1+\tilde r_0)x\in B_{r_0}(x)
$$
and, since $B_{r_0}(x)\subset E^\circ\subset E$, we get that:
$$
(1+\tilde r_0)x\in E\Leftrightarrow x\in \frac{1}{1+\tilde r_0}E
$$
thus
$$
p_E(x)\le\frac{1}{1+\tilde r_0}<1
$$
so $x\in E_1$ and thereby $E^\circ\subset E_1$
