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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, http://users.cms.caltech.edu/~jtropp/notes/Tro00-Cone-Theorem.pdf.2012-08-08
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    I think it is equivalent. Nevertheless, here there is another http://downloads.hindawi.com/journals/fpta/2009/609281.pdf2012-08-08
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    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 $C\cap (-C)=\{ 0\}$?2012-08-08
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    Ah, that was a mistake. True.2012-08-08

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