1. Not finding help here? Sign up for a free 30min tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

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

    User Avatar
    Science Advisor
    Homework Helper

    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

    User Avatar
    Science Advisor
    Homework Helper

    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

    User Avatar
    Science Advisor
    Homework Helper

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

    matt grime

    User Avatar
    Science Advisor
    Homework Helper

    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

    User Avatar
    Science Advisor
    Homework Helper

    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
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?



Similar Discussions: Convex Set
  1. Convex Function (Replies: 2)

  2. Convex Sets (Replies: 1)

Loading...