[EM] Measures of Compactness

[Toby Pereira]

 Minimising average distance is equivalent to using the median, whereas minimising the squared distance is equivalent to the mean. I would say the mean makes more sense. Obviously a contrived example, but if you just had two population points of equal size, minimising the squared distance would put the centre halfway between the points. Using average distance would make any point along the line between the two points equivalent.
Toby
    On Monday 22 September 2025 at 00:16:40 BST, Andrew B Jennings (elections) via Election-Methods <election-methods at lists.electorama.com> wrote:  
 
 I'm a supporter of defining compactness and then using a districting map that optimizes it. Brian Olson (bdistricting.com/about.html) has always advocated that:

> The best district map is the one where people have the lowest average distance to the center of their district.
But what about using the square of the distance? Mathematically, it makes so many square roots disappear that the problem might be much more tractable.
It also seems that minimizing the sum of squares of distances from each voter to the (population-weighted) centroid of their district and minimizing the sum of squares of distances between each pair of voters in the same district is practically the same problem (at least on the plane). That seems nice.
I have some friends who actually want to propose redistricting our city mathematically. Is it worth using the square of the distance?
I have done some maps minimizing the sum of the distances (and squares of distances) between all pairs of voters in the same district. They look pretty similar, at least for my city.
~ Andy Jennings----
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