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A convex set $M$ is called convex body if it has nonempty interior. Interior $I(M)$ consists of elements of $M$ such that $x+ty$ is in $M$ for any $y$ and positive number $r=r(y)$ such that absolute value of $t$ smaller than $r$.

In $\ell^2$, $M$ is the set of sequences such that infinite sum of squares less than or equal to $1$. Show that $M$ is convex but not convex body.

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    You have to specify what vector space you're working in. If it's $\ell_2$, then $M$ is a convex body. If it's some space $V$ that properly contains $\ell_2$, try taking $y$ to be a member of $V$ that is not in $\ell_2$.2012-01-03

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