The Corner

Convexity in Politics

“Mr. Derbyshire—I have been pushing (among my peers) the following

constitutional amendment.

“‘Each state must choose its congressional districts to have the following

properties: 1) Each congressional district has the same population, and 2)

The points lying on straight line between any two points in the district

must, if in the state, also be in that district.’

“The upshot is that congressional districts must either be convex sets, or

if not convex, then the non-convexity is due to the non-convexity of the

state itself. It is easy to prove that there always exists a way to do

this if things are sufficiently smooth.

“Let F be a measure mapping subsets of the state to R. (By smooth, I mean

assume that sets with zero area have zero F-measure.) Next, take a

north-south line, and starting from the western border, move it eastward

until the F-measure of the area west of the line is P/N, where P is the

population of the state and N is the number of districts. Then, starting

from here, do this N-2 more times.

“But this will never get adopted. In whose interest is it to push this?”

Well, it seems to me it is in the national interest, Sir. But then, so is

the flat tax, nuclear power, immigration enforcement,…

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