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.)

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2009-14 New notion on the convexity, 4.9 out of 5 based on 13 ratings

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WHYCyrve의 Similarity의 정의에서 C를 magnifying 해도 좋다는 말이 reducing도 된다는 말을 포함하나요?

S. OumPost authorYes. It is OK to shrink a curve to get a similar curve.

JSGP와 Q를 연결하는 모든 C와 similar한 커브가 S 위에 있어야 하나요?

S. OumPost authorC와 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.

JSG4개가 가능하다면 curve를 뒤집는 것도 되는건가요?

S. OumPost author문제와 큰 상관은 없겠지만 지금 적힌 것만 읽어서는 뒤집어도 무관한 것 같네요.