The total population and the overall average density are derived for a single city, in which the spatial structure conforms approximately to the negative exponential function. It is then shown that across the cities of an urban system, the areal extent of a city is positively related to its average density and also to its population. Consideration is next given to a system of regions and again to the manner in which areal extent is related to average density and population across regions. In this setting, however, the relationships are usually different from those found within an urban system.
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Parr, J.B. Spatial-structure differences between urban and regional systems. Ann Reg Sci 49, 293–303 (2012). https://doi.org/10.1007/s00168-010-0421-5