[EM] Exact spatial model probabilities?

[Daniel Carrera]

On Thu, Jan 27, 2022 at 4:20 AM Kristofer Munsterhjelm <km_elmet at t-online.de>
wrote:

> > Finding the area of a Voronoi cell in a 2D space with just 3-4 cells
> > doesn't sound difficult. I haven't done the math, but it feels like high
> > school math problem.
>
> It isn't very difficult. Fortune's algorithm can get the cells in O(c
> log c) time, and then you can find the area of each (given its vertices)
> by sorting the vertices in a particular order, in O(v log v) time. IIRC,
> the number of vertices of a 2d Voronoi cell is bounded, so that is in
> essence constant time.
>
> A side note: I seem to recall that Patterns of Democracy (by Lijphart)
> says that as a reasonable approximation, the number of dimensions in a
> country's political realm is the number of effective parties, plus one.
> (And anecdotally there definitely seems to be more than one dimension to
> politics here.)
>

You know, if we embrace the idea that the number of dimensions scales with
the number of parties and make it part of the model, that simplifies the
possible shapes of the Voronoi cells:

a) 3 candidates in a 2D space, as long as they don't fall on a line, must
have Voronoi cells that all meet at a point.

b) 3 candidates in a 3D space, as long as they don't fall on a line, must
have Voronoi cells that all meet at a line.

c) 3 candidates in a 4D space, as long as they don't fall on a line, must
have Voronoi cells that all meet on a plane.

More generally, my intuition is that N candidates in a space with at least
N-1 dimensions cannot "surround" each other and that limits the possible
set of shapes their Voronoi cells can have. Furthermore, I suspect that if
the space has exactly N-1 dimensions, the Voronoi cells probably all meet
at a point and the cells look similar to triangles/pyramids/etc. They're
not exactly triangles/pyramids because the one side of the cell ends at the
boundary of the space and we haven't defined what shape that's going to
have. Nonetheless, my point is that we don't need to compute the volume of
a fully generic Voronoi cell in higher dimension; just a relatively simple
subtype.

Cheers,
-- 
Dr. Daniel Carrera
Postdoctoral Research Associate
Iowa State University
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