Kaushik Basu proving himself the worst economist in the world.

Consider the following-

This is simply Hotelling's law. Everybody gravitates to the centroid. It is a Nash equilibrium, even though nobody may be able to locate at the exact center. Even if one vendor locates at the exact center, his rivals set up as close to him as possible because the cost of walking a few extra meters is immaterial to the consumer.  This is Econ 101. Basu disagrees. He posed the question on Twitter and then gave this utterly foolish answer

There is no Nash equilibrium. Proof: If there’s an equilibrium both vendors must be at same point.

No. Both will be as close to the centroid as is possible. Even if one is at the exact center and the other is 20 meters away, consumers will not differentiate between them. The 'marginal cost' of walking an extra 20 meters is zero. 

 Further, that point must be one where every straight line cuts the population in half. (Otherwise one of them will move to the populated side.) 

No. The population experiences zero marginal cost in walking ten or twenty meters if it has travelled many miles to get to the center. Consider what happens when we go to China town.  We enjoy walking around and comparing the menus of different restaurants. There is likely to be 'product differentiation' because what we have here is 'monopolistic competition'. All this is Econ 101. 

Check that no such point exists.

This is foolish. The centroid is the 'Schelling focal' solution to a coordination game. Nobody has to occupy the exact centroid. Being within line of sight of it is good enough. Thus all vendors will try locate centrally. All will fail, because of granularity, but even the unluckiest in this respect, have no incentive to relocate.