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Let $C$ be a convex set with nonempty interior in a topological vector space. Do we always have : $\overline {C^\circ} = \overline{C}$ ?

If not, what is the "minimal" condition on the space so that this equality is true?

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  • $\begingroup$ Cf. Bourbaki, EVT II.2.6, Proposition 16 Corollaire 1. $\endgroup$ – Fred Rohrer Apr 23 '17 at 9:41
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Yes. Assume WLOG that $0$ is an interior point and $y$ is in $C$. Draw a picture to see that the interval $[0,y)$ is contained in the interior of $C$. Specifically, if $U$ is a neighborhood of $0$ contained in $C$ and $0<s<1$, then $sy + (1-s) U \subset C$.

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