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Convex Set

  1. Mar 6, 2005 #1
    Given a convex set X and a convex function f: X - R, show that for any c from R, the set S={x from X: f(x)<=c} is convex

    Any advice about how to prove it?
     
    Last edited: Mar 6, 2005
  2. jcsd
  3. Mar 6, 2005 #2
    The set S is ......?
     
  4. Mar 6, 2005 #3
    Sorry, the set S is convex.
     
  5. Mar 6, 2005 #4
    Convexity

    The set X is convex if for any x, y from X, we have that the line segment joining x and y: ax + (1-a)y, also belongs to X, for any scalar a from (0, d], d > 0.
     
  6. Mar 7, 2005 #5

    matt grime

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    So how many convex subsets of R are there?
     
  7. Mar 7, 2005 #6
    No, just prove that S is a convex set, given the definition of S.
     
  8. Mar 7, 2005 #7

    matt grime

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    Erm, yeah, but it appears obvious. If a and b are in S, then the line segment between them is in X, hence the image of the line segment is a convex subset of R, a and b both satisfy f(a) and f(b) <=c so, I repeat, what does a convex subset of R look like?

    EDIT think i have a different notion of a convex function than you. i'm guessing you mean that f is convex if for each a and b and x any point on the line segment a to b then f(x) < = (f(a)+f(b))/2, but that makes it even easier.
     
    Last edited: Mar 7, 2005
  9. Mar 7, 2005 #8
    Must be like a ball or circle.
     
  10. Mar 7, 2005 #9

    matt grime

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    Check the edited post
     
  11. Mar 7, 2005 #10
    Well, sound like a midpoint of a linesegment
     
  12. Mar 7, 2005 #11

    matt grime

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    what sounds like a midpoint of what linesegment?

    if x is on the line segment from a to be and a and b are in S, then f(x) <= (f(a)+f(b))/2 <= (c+c)/2 = c hence x is S. Thus S is convex.
     
  13. Mar 7, 2005 #12
    Oh, I see. But well, how it looks like graphically? Is a line inside of a circle or something?
     
  14. Mar 7, 2005 #13

    matt grime

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    is what line inside of what circle?
     
  15. Mar 7, 2005 #14
    I think I am losing the point here, sorry about that. So, the point here is that, if S is a convex subset of R, and X is a convex subset of R, and for both of them exists a function f, then for any two points x and y from X, they also belong to S such that S = {x from X: f(x)<=c}.
    In such a way that:
    f(ax+(1-a)y)<=af(x)+(1-a)f(y)
    then
    <=ac+(1-a)c = c
    for which f(x)<=c, this implies that S is convex.
    Right? Sorry if I am wasting you time... =S
     
    Last edited: Mar 7, 2005
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