Let C be a continuous and self-avoiding curve on the plane. A curve from A to B is called a C-segment if it connects points A and B, and is similar to C. A set S of points one the plane is called C-convex if for any two distinct points P and Q in S, all the points on the C-segment from P to Q is contained in S.

Prove that  a bounded C-convex set with at least two points exists if and only if C is a line segment PQ.

(We say that two curves are similar if one is obtainable from the other by rotating, magnifying and translating.)