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!

Generalized triangle inequality

  1. Aug 15, 2012 #1
    1. The problem statement, all variables and given/known data

    Show that

    |x_1 + x_2 + · · · + x_n | ≤ |x_1 | + |x_2 | + · · · + |x_n |

    for any numbers x_1 , x_2 , . . . , x_n


    2. Relevant equations

    |x_1 + x_2| ≤ |x_1| + |x_2| (Triangle inequality)


    3. The attempt at a solution

    I tried using the principle of induction here, but to no avail.

    Can I induct on the basis |x_1| ≤ |x_1| ?
     
  2. jcsd
  3. Aug 15, 2012 #2

    gabbagabbahey

    User Avatar
    Homework Helper
    Gold Member

    Hint: [itex]| x_1+x_2 + x_3 | \leq |x_1 + x_2| + | x_3 |[/itex]
     
  4. Aug 15, 2012 #3
    so I can write |x_1 + x_2 + x_3| ≤ |x_1| + |x_2| + |x_3|

    since |x_1 + x_2| ≤ |x_1| + |x_2|

    but how do I cover all the "n" cases?
     
  5. Aug 15, 2012 #4

    gabbagabbahey

    User Avatar
    Homework Helper
    Gold Member

    Use the inductive principle... assume that [itex]| \sum_{i=1}^n x_i| \leq \sum_{i=1}^n |x_i|[/itex] for some some [itex]n=k[/itex] (it is obviously true for n=1, 2 and 3) , and then show that it must then also be true for [itex]n=k + 1[/itex].
     
  6. Aug 15, 2012 #5

    Ray Vickson

    User Avatar
    Science Advisor
    Homework Helper

    You can use the easily-proven fact that the absolute-value function is convex, in the sense that f(x) satisfies [itex]f(\alpha w_1 + (1-\alpha)w_2) \leq \alpha f(w_1) + (1-\alpha) f(w_2)[/itex] for all [itex] \alpha \in [0,1].[/itex] Try to prove that
    [tex] \left| \frac{x_1 + x_2 + \cdots + x_n}{n}\right| \leq \frac{1}{n}|x_1| + \cdots + \frac{1}{n} |x_n|.[/tex] Hint: induction.

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




Similar Discussions: Generalized triangle inequality
  1. Triangle Inequality (Replies: 7)

  2. Triangle inequality (Replies: 10)

  3. Triangle inequality (Replies: 2)

  4. Triangle Inequality (Replies: 2)

Loading...