I was curious how many people lived in a roughly five square mile area of downtown and it's immediate area for selected cities. I tried to get as close to five square mile as possible while:
1. including the entire area that would be referred to by locals as "downtown" (if I'm familiar with the city)
2. trying to get as close to a roughly rectilinear area as possible, while still counting only entire census tracts.
Since the areas are approximate, I've rounded both the areas and the population densities to two significant digits. The numbers are all from the 2010 U.S. Census, and the 2011 Canadian Census.
The Montreal number is far lower than I had anticipated. It's possible I missed some census tracts or miscalculated when I was adding them up.
I was going to add some more cities when I have time. Here are the cities I've done ordered by average density:
1. New York
population: 370,430
approx. area: 4.9 square miles
approx. density: 76,000 ppsm
2. Toronto
population: 175,064
approx. area: 5.0 square miles
approx. density: 35,000 ppsm
3. Vancouver
population: 163,753
approx. area: 5.3 square miles
approx. density: 31,000 ppsm
4. San Francisco
population: 128,317
approx. area: 5.0 square miles
approx. density: 26,000 ppsm
5. Chicago
population: 130,308
approx. area: 5.3 square miles
approx. density: 25,000 ppsm
6. Montreal
population: 89,636
approx. area: 4.7 square miles
approx. density: 19,000 ppsm
7. Seattle
population: 80,438
approx. area: 4.4 square miles
approx. density: 18,000 ppsm
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