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.)
Cyrve의 Similarity의 정의에서 C를 magnifying 해도 좋다는 말이 reducing도 된다는 말을 포함하나요?
Yes. It is OK to shrink a curve to get a similar curve.
P와 Q를 연결하는 모든 C와 similar한 커브가 S 위에 있어야 하나요?
C와 similar한 모든 curve가 S 위에 있어야 겠죠. (많아야 4개 밖에 없지 않나요? 시작점 끝점이 고정되니까요)
Yes.. All curves from P to Q similar to C should be on S.
4개가 가능하다면 curve를 뒤집는 것도 되는건가요?
문제와 큰 상관은 없겠지만 지금 적힌 것만 읽어서는 뒤집어도 무관한 것 같네요.