1. Limited time only! Sign up for a free 30min personal 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!

Homework Help: Analysis Q: Convex subsets

  1. Feb 29, 2008 #1
    1. The problem statement, all variables and given/known data

    Let U be a non-empty, convex, open subset of R^2. Prove that U is homeomorphic to R^2.
    Hint: First prove that the intersection of a line in R^2 with U (if non-empty) is homeomorphic to an open interval in R^1. Then use radial projections.

    2. Relevant equations

    We just have the basic definition of homeomorphism and some standard results about open/closed sets to work with. And of course the completeness axiom for R.

    3. The attempt at a solution

    Ok well, I have proven that the intersection of a line in R^2 with U (if non-empty) is homeomorphic to an open interval of R^1. I see the idea of the proof is to translate U homeomorphically so that the origin is at an interior point of U, and then radiate lines outward from the origin in all directions (think: polar coordinates). Since U is convex, the lines from the origin to the "boundary" of U are fully contained in U and comprise all of U; Since U is open, such lines to the "boundary" can be mapped to the FULL lines extending forever outward in the corresponding direction in R^2.

    So I can make a bijection from such projected "lines" in U to all of R^2; in fact for the case where U is bounded I've made an explicit bijection which I suspect is both-ways continuous but I'm having trouble proving it.

    The notation I'm using is

    R(theta) = supremum of the distances of points of the "line" in U which points in the direction of theta. (ie, R(theta) is the distance from the origin to the "boundary" of U along the direction of theta)

    I believe the following bijection sends a "line" in U in the direction of theta to the full line in R^2 in the direction of theta:

    f(r) = r/ [R(Theta)- r]

    where r is interpreted as distances along the direction of theta in U.

    If I could prove that R(theta) is a continuous function of theta then I think I would know how to proceed. It seems obviously true because a jump discontinuity in the boundary of a convex space would seem to produce a contradiction to convexity. But that's just for the bounded case!

    In any case, I am stumped, and I'm thinking there must be a better way to do this. Is there a better approach? Thanks guys.
    Last edited: Feb 29, 2008
  2. jcsd
  3. Mar 1, 2008 #2
    any ideas?
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook