Looking for an algebra of vertices...

[phaedo]

Hello,

 

My knowledge of abstract algebra beyond linear vector spaces is very limited. My problem is inspired from the (not very well-known) triangular inequality between angles of a tetrahedron (see for example http://convexoptimization.com/wikimization/index.php/Fifth_Property_of_the_Euclidean_Metric):

[latex]

\left| \widehat{x,y}-\widehat{y,z} \right| \leq \widehat{x,z}\leq\widehat{x,y}+\widehat{y,z}

[/latex]

where x,y,z are three vectors and [latex]\widehat{x,y}[/latex] is the angle formed by the vectors x,y.

 

I am looking for a way to abstract this into some algebra of vertices. Say a vertex is the tuple [x,y], we would need to define some "addition" operator in a transitive way so that [x,y]+[y,z] = [x,z], and we would define a norm as [latex]\| {[x,y]} \| = \widehat{x,y}[/latex]. The triangular inequality above would then read in the familiar way:

[latex]

\left| \| [x,y] \| - \| [y,z] \| \right| \leq \| [x,y] + [y,z] \| \leq \| [x,y] \| + \| [y,z] \|

[/latex]

 

Does this look familiar to anyone?

 

Thanks in advance for any pointers.

 

p.

[ajb]

[x,y] +[y,z] = [x,z] reminds me of the partial multiplication found in groupoids. You want [x,y]+ [y', z] = [x,z] only if y = y' and leave it undefined otherwise?

[phaedo]

[x,y] +[y,z] = [x,z] reminds me of the partial multiplication found in groupoids. You want [x,y]+ [y', z] = [x,z] only if y = y' and leave it undefined otherwise?

 

Thanks for pointing groupoids out. It would be nice to find a generalization of addition for any [x, y] + [z, u], but it's not required.

 

One difficulty is that the norm I am considering here is bounded... Not sure if we can make sense of this.

[ajb]

One difficulty is that the norm I am considering here is bounded... Not sure if we can make sense of this.

Why is having a bounded norm a problem?

 

I do not see that being bounded or not is part of the definition of a norm. You should check that the norm you define really is a norm, that is satisfies the properties you need it to satisfy.

 

i) Absolute homogeneity

ii) The triangle inequality

iii) Separates points (norm of the zero vector is zero)

[phaedo]

If norm(a.x) = | a | norm(x) then as | a | goes to infinity the norm of a.x goes to infinity... Thus a norm can't be bounded?

[ajb]

Okay, the absolute homogeneity rules bounded out. Good point.

[Keen]

What exactly is an algebra of vertices to you? It seems to me that what you are looking for is simply an affine space.