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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.

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Are you sure you have the definition of a cone correct? See, for example, – Jonathan Gleason Aug 8 '12 at 22:14
I think it is equivalent. Nevertheless, here there is another – Michael's selection Aug 8 '12 at 22:46
Suppose $C$ contains two distinct $v_1$ and $v_2$. Then, according to your definition, $v_1-v_2=0$, i.e. $v_1=v_2$. Thus, the only $C$ that satisfy your definition are singletons. Are you sure you didn't mean $C\cap (-C)=\{ 0\}$? – Jonathan Gleason Aug 8 '12 at 22:51
Ah, that was a mistake. True. – Michael's selection Aug 8 '12 at 23:17

Counterexample: let $C$ be the quadrant $x,y\ge0$ in the plane. Let $e=(1,0)$. The set $C-e$ does not contain the origin as an interior point.

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