A property extending convexity

Question

Recently, I was trying to prove the simply connectedness of some set $Σ$, finally I've found that for any two points in $Σ$ there is a path in Σ joining them consisting of the union of two segments. This looks a bit like convexity, so I'm wondering if this property is already known under some designation or if it has other topological consequences and interpretations. Thank you !