Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Metric function composed with concave function

  1. Jan 9, 2012 #1
    Hi,

    I have been reading about metric spaces and came across an elementary property that I am having difficulty proving. A quick search on these forums and google has also failed.

    Given a metric space with distance function [itex]d[/itex], and an increasing, concave function [itex]f:\mathbb{R} \rightarrow \mathbb{R}[/itex] so that [itex]f(0)=0[/itex], show that [itex]f\circ d[/itex] is a metric.

    Of course, only the triangle inequality is nontrivial.
     
  2. jcsd
  3. Jan 11, 2012 #2

    morphism

    User Avatar
    Science Advisor
    Homework Helper

    Hint: First show that f(tx) >= tf(x).
     
  4. Jan 20, 2012 #3
    I have been struggling with this problem all day so I described it in a google search and found this forum.

    I have that f(d(x,y)) <= f(d(x,z)+d(z,y)) but I hit a brick wall when I try to "free" the d's out of the function, i.e. I get for example that f(d(x,z)+d(z,y)) >= (d(x,z)+d(z,y))*f(1) =d(x,z)*f(1) + d(z,y) * f(1) <=f(d(x,z)+f(d(z,y)), but that's worthless because the inequalities go back and forth.

    I also tried putting f(d(x,z)+d(z,y)) = f((a+b)(d(x,z)+d(z,y))) = f(a*(d(x,z)+d(z,y))+b*(d(x,z)+d(z,y))) >= f(a*d(x,z)+b*d(z,y)) >= a*f(d(x,z)) + b*f(d(z,y)) but that doesn't give me anything useful.

    Can anyone give another tip how I should be thinking about this problem?
     
  5. Jan 20, 2012 #4

    micromass

    User Avatar
    Staff Emeritus
    Science Advisor
    Education Advisor
    2016 Award

    Did you first show that mathman's hint is correct??
     
  6. Jan 21, 2012 #5
    Yes, I put y=0 in the equation f(ax + by) >= af(x) + bf(y), and in my calculations I tried to apply the hint but it didn't get me anywhere.
     
  7. Jan 21, 2012 #6

    micromass

    User Avatar
    Staff Emeritus
    Science Advisor
    Education Advisor
    2016 Award

    Now write

    [tex]f(a)+f(b)=f\left((a+b)\frac{a}{a+b}\right)+f\left((a+b)\frac{b}{a+b}\right)[/tex]

    Apply the hint with t = the fractions.
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook