Yes, convexity in the geometrical sense is a bit hard to define. Here’s one way to imagine it:

Suppose that we have six seats to fill. We take our map of the state and plunk down six dots randomly but well-spaced. Then imagine a kind of two-dimensional rubber bubble placed on each dot. We start pumping air into each of the bubbles. Where two bubbles meet, they form a linear boundary, but they continue to expand into open territory. Continue pumping in air until the map is completely covered by the bubbles.

There are definitely some tricky issues. This scheme covers territory neatly, but not population. To make them equal by population, you have to alter the amount of “air” that you pump into each bubble as it gets more people inside it. It’s also vulnerable to initial conditions: if the initial placement of the dots is way off, you can end up with one bubble squeezing between two other bubbles and expanding into territory on the other side of the “pass” between the two bubbles — it would look gerrymandered. You get around this by trying a zillion initial starting points for the dots, and seeing what you get. You can calculate the convexity by measuring the lengths of the boundary lines. Out of the zillion maps you make, measure the lengths of the boundary lines and choose the map with the lowest total length.

This scheme ends up producing mostly distorted hexagons. This lines up neatly with a theorem from geometry proving that the hexagon is the most-sided polygon with which you can “tile” a plane.

Yep, definitely too mathematically hairy for almost anybody! 😁

Master of Science, Physics, 1975. Computer Game Designer. Interactive Storytelling.

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