More generally, if is a normed vector space then any connected open subset of is path connected.

This follows from the fact that ever locally path connected and connected space is path connected. But, we know that is connected and since every point of has some open ball containing it sitting inside , but since is convex and thus path connected you have that is locally path connected.

If you want to learn more, or see proofs of the above you can look at my blog post here (sorry, the formatting on that one is pretty bad).