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Math Help - Convexity

  1. #1
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    Convexity

    Let x and y be distinct points in a Euclidean metric space. The set \lbrace z \vert d(z,x) < d(z,y)\rbrace is convex. Does the same apply for any linear combination of the distances, i.e.: Is \lbrace z \vert \lambda d(z,x) < (1 - \lambda) d(z,y)\rbrace, where 0 \leq \lambda \leq 1, in any case convex?
    Last edited by Richard; February 12th 2010 at 01:17 AM.
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  2. #2
    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by Richard View Post
    Let x and y be distinct points in a Euclidean metric space. The set \lbrace z \vert d(z,x) < d(z,y)\rbrace is convex. Does the same apply for any linear combination of the distances, i.e.: Is \lbrace z \vert \lambda d(z,x) < (1 - \lambda) d(z,y)\rbrace, where 0 \leq \lambda \leq 1, in any case convex?
    I just did this recently on another site . What have you tried?
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  3. #3
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    Convexity

    It's clear to me that for the simple case that \lambda = 1 - \lambda = .5, the set must be convex. To generate the said set, you just need to draw a line through the two points, half it, and draw a perpendicular line through the half. The halfplane on the side of x is then the said set, which is convex. But I did not manage to generalise the proof.
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