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Re: Struggling to prove simple triangle inequality

• To: mathgroup at smc.vnet.net
• Subject: [mg126339] Re: Struggling to prove simple triangle inequality
• From: Murray Eisenberg <murray at math.umass.edu>
• Date: Wed, 2 May 2012 05:45:26 -0400 (EDT)
• Delivered-to: l-mathgroup@mail-archive0.wolfram.com
• References: <jnljcu$n5b$1@smc.vnet.net> <201205011858.OAA10792@smc.vnet.net>
• Reply-to: murray at math.umass.edu

On my system (OS X), Timing[Reduce[inequalityb, vars, Reals]] gives 
1.0295. But if you try instead to prove the Cauchy-Schwartz Inequality, 
which is equivalent to the Triangle Inequality ...

   Reduce[Abs[a.b]^2 <= (a.a) (b.b), Variables[{a, b}], Reals] // Timing

... the timing is 0.18544, which is 10-fold faster!

On 5/1/12 2:58 PM, danl at wolfram.com wrote:
> On Monday, April 30, 2012 3:44:14 AM UTC-5, Vladimir M wrote:
>>
>> ...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 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.
>
> 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}

-- 
Murray Eisenberg                     murray at math.umass.edu
Mathematics & Statistics Dept.
Lederle Graduate Research Tower      phone 413 549-1020 (H)
University of Massachusetts                413 545-2859 (W)
710 North Pleasant Street            fax   413 545-1801
Amherst, MA 01003-9305

• References:
  • Re: Struggling to prove simple triangle inequality
    • From: danl@wolfram.com