is a tribute to Thales, since so much of it is about the geometry of circles. In the next section it will be shown how the fact that people live all over the globe can be transformed to a mathematically equivalent problem in which people live all over a circle; and then, using this I will demonstrate some results pertaining to the globe.

The other key figure for this paper is John Nash (1928‐2015). Born on June 13, in Bluefield, West Virginia, Nash made important scientific contributions to numerous fields, including differential geometry, singularity theory, and most importantly game theory and the creation of a concept of equilibrium that is central to contemporary economic analysis. His achievements are all the more remarkable because schizophrenia claimed him by the time he was 30 years old. So all his staggering achievements had to be squeezed into six or seven crowded years. In keeping with this remarkable brevity was his PhD thesis in mathematics from Princeton University—just 28 pages long. A personal reason for me to want to give a tribute to Nash is an encounter with him in 2003. This was at a conference on game theory in Mumbai, partly to honor John Nash, who had by then won the Nobel Prize (1994), and his schizophrenia was in some remission, though he was still clearly distracted and unmindful. As I began presenting my paper, I was touched to see Nash walk in and sit down in the front row. I was nervous at the thought of speaking of Nash equilibria in Nash’s presence. But I need not have been, for he fell asleep within the first five minutes and did not wake up till people clapped at the lecture’s end.

# **3. Spheres and Circles**

The location problem described in the opening section has people staying all over a globe or a sphere and buying from the nearest store located on a rotary or a circle. The first result I want to establish is an equivalence. If the people, instead of being uniformly distributed on the sphere, were instead uniformly distributed on the rotary or circle where the stores are located, with the rest of the sphere uninhabited, the mathematical problem would be exactly the same. Indeed, there is a general principle regarding how any distribution (not necessarily uniform) on a sphere can be converted to an equivalent distribution on a circle and then for the analysis to be done on a circle.

To establish this, I first need a sphere. Fortunately, we have a readymade one. The World Bank Group logo turns out to be just what we need for this exercise. The logo is reproduced in Figure1. Suppose people live all over this sphere and the rotary on which all stores are to be located happens to be the circle going through A, J, and B.

What I will first establish is an easy way to locate how a person living anywhere on the sphere will choose which store to go to (recall people go to the nearest store). To do this, label the center of the rotary (on the sphere) as N. This is as shown in Figure 1. I shall refer to this as the ‘relative North Pole,’ that is, relative to the circle. Mark the opposite point (not shown in the figure) as S, the relative South Pole. Next draw the latitude and longitude lines on the sphere with reference to these two poles, N and S. These will be called the ‘relative’ latitude and longitude lines. They are the lines drawn relative to the starting circle. If the starting circle was elsewhere, the relative latitude and longitude lines would, typically, be elsewhere. Luckily, the World Bank logo comes with these latitude and longitude lines marked. These lines are on display in Figure 1.

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