[klhwfczr] McSparseness scaled by density

Van Worley's McSparseness calculation found that the place farthest from a McDonalds is in some sparsely populated region.  This is not surprising.

Let's incorporate population density, assumed to be a continuous function.

First way: create a Voronoi cell around each restaurant.  Which cell has the greatest population?

Second way: Consider a point x.  Draw a straight line from x to its nearest restaurant, and compute the path integral of population density along that path: How many people will you walk by on the way to the geographically nearest McDonalds?  Which point maximizes this value?

Third way: instead of a straight path, choose, among all possible paths, the one that minimizes the path integral.  This is similar to light refracting through a medium with variable index of refractivity.  This avoids the weird case where a straight line path happens to cross over a small patch of very high density.  The minimum path might not be the one to the geographically closest restaurant.  What point maximizes the minimum path? (Minimax)

Let the density function be population density plus a constant.  This becomes a mix of distance and density.  As this constant varies, how does the maximum point change?