Let $C$ a subset of a normed vector space $V$.
Then it is straightforward that :
$C$ is convex $\Rightarrow$ $C$ is connected,
and this result is cited everywhere.
But it is also true that :
$C$ is convex $\Rightarrow$ $C$ is locally connected.
Here is the proof.
Suppose $C$ is convex, let $x \in C$ (if $C = \varnothing$ the statement is of course true), and $U$ an open set of $C$ containing $x$. We can write $U = U' \cap C$ where $U'$ is an open set of $V$. Since $U$ contains $x$, $U'$ contains $x$. Therefore there exists $r > 0$ such that $B(x,r) \subseteq U'$. Then $B(x,r) \cap C \subseteq U$. $B(x,r) \cap C$ is a convex set (as an intersection of two convex sets), open in $C$, and contains $x$, so it does the job. Hence $C$ is locally connected.
However, there seems to be no reference about the second result on the internet. The Wikipedia article on locally connected spaces doesn't mention it.
Two questions :
- Is the result or my proof wrong (which I highly doubt) ?
- Does someone know about a reference that mentions the result ?
Your proof looks correct to me.
I don't know of any reference, but if there is one it will be some kind of textbook. Honestly, the result is really easy to prove, so there will certainly not be a paper dedicated to it. Maybe some articles use this, but I don't think you will find any kind of "standard reference" for this result.