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Math Help - Convex set proof

  1. #1
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    Convex set proof

    I have a question about convex set. how can I prove that a set is convex if and only if its intersection with any line is convex. I have worked out proof but it is not that rigorous and I was wondering if someone could help me with this
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  2. #2
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    Quote Originally Posted by ROOZ1234 View Post
    I have a question about convex set. how can I prove that a set is convex if and only if its intersection with any line is convex. I have worked out proof but it is not that rigorous and I was wondering if someone could help me with this
    Why not post what you have done? We can look at it then.
    Here is a hint: Two points determine a unique line and lines are convex sets.
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  3. #3
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    if S is convex
    exist x,y that belongs to S -> lambda(x) + (1-lambda) (y) belongs to S
    therefore x and y are on the line L
    ->lambda(x) + (1-lambda) (y) belongs to L
    ->lambda(x) + (1-lambda) (y) belongs to intersection of L and S
    therefore S interesection L is convex

    there exists x,y that belongs to S intersection with L -> lambda(x) + (1-lambda) (y) belongs to S intersection with L

    -> lambda(x) + (1-lambda) (y) belong to S
    ->S is convex
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  4. #4
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    You certainly have all to components there.
    I do not know the level of rigor required of you.
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  5. #5
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    thx for your help
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