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,

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

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