# [EM] Advantages of Dirichlet Region Districts

Forest Simmons fsimmons at pcc.edu
Wed Jan 14 14:30:29 PST 2004

```If voting districts were required to be Dirichlet regions (with respect to
their "centers"), then ...

... you could submit a redistricting proposal on a postcard by just
listing the coordinates (latitude and longitude) or addresses of the
proposed district "centers."

... it would be easy to determine which voter addresses belonged to which
districts without ever drawing the district boundaries. (MapQuest software
will do it.)

... consequently it would be easy to detect which proposals failed to
satisfy the equality tolerance.

... if a (literal) taxicab metric were adopted, then districts would
automatically tend to respect natural boundaries like rivers, parks, and
other areas of relatively sparse voter population.

... if (among all of the proposals passing equality muster) the one with
the least average distance from voter to district "center" were chosen
(according to law), then the regions would tend to be almost as compact as
possible.

Related Remark:

and find out which of the ten thousand plus LDS chapels is nearest your
home.  Then (by clicking correctly) you can get a MapQuest map of the
driving ("literal taxicab metric") distance indicated.

Forest

On Mon, 12 Jan 2004, Forest Simmons wrote:

> I suggest (as one of the constraints) that districts should be Dirichlet
> regions with respect to some set of defining points, P_1, P_2, P_3, ... ,
> P_n, one defining point for each of the n regions.
>
> In other words if you are in district k, then point P_k should be closer
> to your residence than any of the other defining points.
>
> If the defining points are the precinct headquarters, then you just vote
> at the precinct closest to your home, and you will automatically get the
> correct ballot.
>
> If distance is measured with the Euclidean metric, then the districts will
> be convex polygons called Voronoi polygons.
>
> A more practical metric would be the "estimated travel time" metric.
>
> Rental cars with GPS systems have software for deciding the estimated
> travel time from point A to point B, so we know that this metric can be
> standardized and computed effectively.
>
> This Dirichlet region constraint still leaves enough freedom that other
> requirements can be met, including the relative equality of the number of
> voters in each district.
>
> Minimization of some other quantity (such as total or max travel distance
> for voters) subject to these basic constraints could help select from
> several proposed configurations.
>
> Just an idea (capable of modification or adaptation)!
>
> Forest
>
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>

```