Incidentally, there is actually a mathematical theory of convexity. Prof.
C.A. Rogers, one of my teachers at University College, London, was an
authority on it. It doesn’t seem to be well represented on the web — I
can’t find a neat summary. If you want to try for yourself, good additional
Google arguments are “steiner symmetrization” and “isoperimetric
inequality”.
I have often wondered whether this branch of math might have some
application in politics. We read a lot about the convoluted electoral
districts produced by computer-selecting precincts to make every
congressional seat a safe one. Perhaps that kind of thing could be stopped
if some convexity requirement was placed on the shape of congressional
districts…