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In a problem I'm working on I found myself with a point $y\in \mathbb{R}^m$ lying at the boundary of a non-closed convex set $K$. I'd like to express it as as "infinite convex combination" $y=\sum_{i=1}^\infty \lambda_i y_i,$ where $\lambda_i \ge 0$, $\sum_i \lambda_i=1$ and $y_i \in K$ for all indices $i$. May I do so?

Thank you.

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    You are welcome. Sorry it is not true.2012-11-11

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No. The following is from Convex series (G.J.O. Jameson).

Definition: A set is CS-closed if it contains the sum of every convergent convex series of its elements.

Example (i): An open convex set, $A$, is CS-closed, since $\lambda A+(1-\lambda)\overline{A}\subseteq A$ for $0<\lambda <1$.

This holds for any Hausdoff topological linear space.