[EM] Boundaries between victory regions

[Alex Small]

Greetings, everyone!  I do continue to read this list, even though I don't post, and on the side I continue to work on some theorems related to FBC.  The final step in proving one of the theorems requires some linear algebra.  I'm hoping somebody can help.



Suppose we want to define n distinct,
non-overlapping regions in a vector space of some unspecified dimension greater
than or equal to n.  The boundaries are
defined by linear equations of the form:

 

(eq 1)  (Nij, v) = 0

 

Where (Nij, v) is the inner product between a vector v in the space
and the vector Nij normal to the boundary between regions i and j, pointing
from the boundary into region i.  (So Nji
= -Nij.)  If (Nij,v)>0 then v is on
the i side of the i-j boundary, and if (Nij, v) < 0 then v is on the j side
of the i-j boundary.

 

The normal vectors Nij and Nik for two different boundaries are
related by symmetric, and orthogonal matrices Sjk satisfying:

 

(eq 2)  Nij = Sjk Nik 

 

(eq 3)  Sjk = Skj, Sjk^2 = 1
where 1 is the identity matrix.

 

We require that every point in the vector space either lie on one of
the boundaries or lie in one of the regions. 
In general, the conditions (Nij, v) could specify n(n-1)/2 different
surfaces dividing the space into 2^n regions (plus boundaries) but we are
stipulating only n distinct regions.

 

Does it follow that the normal vectors can ALWAYS be written as:

 

(eq 4)  Nij = Vi-Vj

 

for all i and j, where Vi and Vj are vectors?

 

Clearly the vectors defined in eq. 4 satisfy eq. 2.  They are not linearly independent, so v
having positive projection onto N12 and N23 means that v also has positive
projection onto N13 = N12 + N23.

 

So I know that vectors of the form specified in equation 4 satisfy
all of the conditions sketched out above, but I don’t know if the conditions
sketched out above REQUIRE that Nij satisfy 4.


      
-------------- next part --------------
An HTML attachment was scrubbed...
URL: <http://lists.electorama.com/pipermail/election-methods-electorama.com/attachments/20090724/f3685f41/attachment-0002.htm>