Let $X$ be a Banach space. Let $C\subset X$ be a closed convex cone with nonempty interior. We denote by $B_X$ and $S_X$ the unit ball and the unit sphere of $X$ respectively. Let $e\in C\cap S_X$, $\mathcal{E}>0$ and $\delta>0$. Then, I have to prove that
$\delta \mathcal{E} B_X\subset C-\delta e.$
Note 1: $A-x:=\{a-x\,:\, a\in A\}$.
Note 2: A cone is a set that $\lambda C+\mu C=C$ for every $\lambda,\mu>0$ and $C\cap(-C)=\{0\}$.
Thanks in advance.