2009-14 New notion on the convexity

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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6 thoughts on “2009-14 New notion on the convexity

  1. WHY

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

  2. S. Oum Post author

    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.

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