Mathematica 9 is now available
Services & Resources / Wolfram Forums / MathGroup Archive
-----

MathGroup Archive 2012

[Date Index] [Thread Index] [Author Index]

Search the Archive

Re: Struggling to prove simple triangle inequality

  • To: mathgroup at smc.vnet.net
  • Subject: [mg126335] Re: Struggling to prove simple triangle inequality
  • From: danl at wolfram.com
  • Date: Tue, 1 May 2012 14:58:55 -0400 (EDT)
  • Delivered-to: l-mathgroup@mail-archive0.wolfram.com
  • References: <jnljcu$n5b$1@smc.vnet.net>

On Monday, April 30, 2012 3:44:14 AM UTC-5, Vladimir M wrote:
> Greetings!
> 
> Given two 3D vectors A and B, I want to prove that length of their sum
> is less or equal than the sum of their lengths:
> 
> length[v_] := Sqrt[v.v];
> a = {ax, ay, az};
> b = {bx, by, bz};
> inequality = length[a + b] <= length[a] + length[b];
> 
> This is famous, well-known and quite obvious: triangle side is shorter
> than the sum of other sides, straight line is shorter than non-
> straight, etc. However, proving it formally is hard.
> 
> This fails:
> 
> assum = Element[ax | ay | az | bx | by | bz, Reals];
> FullSimplify[inequality, assum]
> 
> 
> This takes ages on a high-end PC with unknown result:
> 
> vars = {ax, ay, az, bx, by, bz};
> Reduce[inequality, vars, Reals]
> 
> Anyone can help? I think Reduce should somehow make it because it
> actually succeeds at least with 2D vectors.
> 
> 
> --
> All the best,
> Vladimir

You can make it easier by squaring to remove many of the radicals.

In[305]:= length[v_] := Sqrt[v.v];
a = {ax, ay, az};
b = {bx, by, bz};
inequalityb = Expand[length[a + b]^2 - (length[a] + length[b])^2] <= 0;
vars = Variables[{a, b}];
Timing[Reduce[inequalityb, vars, Reals]]

Out[310]= {2.98, True}

Daniel Lichtblau
Wolfram Research



  • Prev by Date: Re: Struggling to prove simple triangle inequality
  • Next by Date: Re: Importing large file into table and calculating takes a long time. How to improve efficiency?
  • Previous by thread: Re: Struggling to prove simple triangle inequality
  • Next by thread: Re: Struggling to prove simple triangle inequality