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Math Help - are these sets are convex or not?

  1. #1
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    are these sets are convex or not?

    how can we determine the following sets are convex or not? can it be enough to draw a 3d graph? or how can we apply the formula f( \alphax + (1- \alpha)y) \leq \alphaf(x) + (1- \alpha)f(y) for f(x1,x2)?


    Which of the following sets are convex and which are not?
    a. {(x1 x2) : x1^2+ x2^2 \leq 1}
    b. {(x1, x2) : x1 = 1, |x2| \leq 4)
    c. {(x1, x2) : x2 — x1^2 = 0}
    d. {(0,0), (0,1), (1,0), (1,1)}
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  2. #2
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    Quote Originally Posted by spring View Post
    how can we determine the following sets are convex or not? can it be enough to draw a 3d graph? or how can we apply the formula f( \alphax + (1- \alpha)y) \leq \alphaf(x) + (1- \alpha)f(y) for f(x1,x2)?


    Which of the following sets are convex and which are not?
    a. {(x1 x2) : x1^2+ x2^2 \leq 1}
    b. {(x1, x2) : x1 = 1, |x2| \leq 4)
    c. {(x1, x2) : x2 — x1^2 = 0}
    d. {(0,0), (0,1), (1,0), (1,1)}
    What 3d graphs? Which function f? As far as I can tell you're asking about convex sets not convex functions, so to see that a set is convex you pick any two points x,y in the set and check that the segment tx+(1-t)y is in the set for all 0<t<1. For the first one it's easier to do if you know that intersection of convex sets is convex.
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