Let $E$ be a Banach algebra, and $e_0\in E$, $e_{0}\neq0$. Suppose $0<r<\|e_{0}\|\leq 1$. Let $E_{1}=\{e\in E:\|e\|=1\}$. Clearly, $e_{0}\notin rE_{1}$. Consider the set $A=\overline{\text{Aco}\{rE_{1}\cup \{e_{0}\}\}}$.
Here $\text{Aco}(X)$ denotes the absolutely convex hull of the set $X$, i.e., $\text{Aco}(X)=\{\sum_{i=1}^{n}\lambda_{i}x_{i}:x_{i}\in X, \lambda_{i}\in \mathbb{K}, \sum_{i-1}^{n}|\lambda_{i}|\leq 1\}$.
My question is the following: what does $\overline{\text{Aco}(A)}$ look like? Is it equal to $\{e\in E:\|e\|\leq\|e_{0}\|$}? This appears to be the case in $\mathbb{R}$ or $\mathbb{R}^{2}$.
I'd be grateful for a hint on this. Thanks in advance.