Convexity of Open Domains

This is probably trivial but for quite a few theorems dom(f) needs to be convex. However, often dom(f) is taken to be open. I know what these signify individually but was wondering if an open dom(f) is also considered convex or whether I would need to prove this as an additonal step to then use another theorem that requires dom(f) to be convex.

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Not all open domains are convex. Take for example the subset of R^2 excluding the (closed) unit disk with radius <= 1. This domain is open, but not convex.

In principle, if a theorem requires a convex domain, you would have to show it, unless it's obvious (e.g. for all real numbers or a higher-dimensional Euclidian space)

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