Polygonally connected set
A set where any two points can be joined by finitely many straight segments that stay inside it.
A set S is polygonally connected if any two points of S can be joined by a polygonal path -- finitely many straight segments laid end to end, each one staying inside S -- even when no single straight segment will do. A convex set is automatically polygonally connected, since one segment is a polygonal path with a single link. The useful direction runs the other way -- for an open subset of the complex plane, connected and polygonally connected turn out to be the same property -- and this is the tool behind carrying a fact from one point of a domain to any other point, one straight segment at a time.
Examples
A punctured disk 0<|z-z_0|<r is not convex but is polygonally connected -- route two segments around the missing center instead of one straight line through it.
An open, connected L-shaped region is polygonally connected -- two segments meeting at the corner join any two of its points.
See also