It is easy to see that every convex set is path connected. What are some examples so that converse holds (not counting the (trivial) one dimensional case)? Is there a nice topology so that this holds?
2026-03-26 17:27:56.1774546076
When does path connectedness implies convexity?
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If the space is such that only constant functions are continuous (space with trivial topology), then the only path-connected sets are singletons, which are convex.