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