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Math Help - Question about convex set

  1. #1
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    Question about convex set

    So i know the definition of a convex set:
    Set S is convex then tx+(1-t)y belongs to S for all 0<t<1

    I have to decide whether the following set is convex:
    {(x,y); x-y<=1}

    I started with tx+(1-t)y = t(x-y)+y <= t+y

    then I am stuck.

    edit: hm I think I might have been mixing up the the points that I choose (x,y) with the variabels in x-y<=1...

    I redid it with matrices and this is what i came up with:

    (1 -1)(x1,x2) <= 1 , c = (1 -1)
    cx <= 1
    cy <= 1

    => c(tx + (1-t)y) = tcx+(1-t)cy <= t + 1 -t = 1

    Is this the correct way to solve the problem?
    Last edited by MagisterMan; November 13th 2011 at 07:53 AM.
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  2. #2
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    Re: Question about convex set

    Quote Originally Posted by MagisterMan View Post
    So i know the definition of a convex set:
    Set S is convex then tx+(1-t)y belongs to S for all 0<t<1

    I have to decide whether the following set is convex:
    {(x,y); x-y<=1}

    I started with tx+(1-t)y = t(x-y)+y <= t+y

    then I am stuck.

    edit: hm I think I might have been mixing up the the points that I choose (x,y) with the variables in x-y<=1... Yes, that is exactly what you were doing!
    In this problem, the set S = \{(x,y): x-y\leqslant1\} is a set of points in two-dimensional space. The definition of convexity says that you have to take two points in S. Call them (x_1,y_1) and (x_2,y_2). The fact that these points are in S tells you that x_1-y_1\leqslant1 and x_2-y_2\leqslant1.

    To see whether S is convex, you have to check whether the point

    t(x_1,y_1) + (1-t)(x_2,y_2) = \bigl(tx_1+(1-t)x_2,ty_1+(1-t)y_2\bigr)

    is in S. The condition for that is \bigl(tx_1+(1-t)x_2\bigr) - \bigl(ty_1+(1-t)y_2\bigr) \leqslant1. So you need to check whether that condition follows from what you are given.
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  3. #3
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    Re: Question about convex set

    Super helpful! Thanks
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