Relatively convex subsets of simply connected planar sets

Let D ⊂ R² be simply connected. A subset K ⊂ D is relatively convex if a, b K, [a, b] K ⊂ D implies [a, b] K ⊂ K. We establish the following version of Helly's Topological Theorem: If K is a family of (at least 3) compact, polygonally connected and relatively convex subsets of D, then ∩ K ≠ 0, provided each three members of K meet. We also prove other results related to the combinatorial metric ρ_K(a, b) (= smallest number of edges of a polygonal path from a to b in K).