  The spatial distribution of income shapes the structure and organisation of
cities and its understanding has broad societal implications. Despite an
abundant literature, many issues remain unclear. In particular, all definitions
of segregation are implicitely tied to a single indicator, usually rely on an
ambiguous definition of income classes, without any consensus on how to define
neighbourhoods and to deal with the polycentric organization of large cities.
In this paper, we address all these questions within a unique conceptual
framework. We avoid the challenge of providing a direct definition of
segregation and instead start from a definition of what segregation is not.
This naturally leads to the measure of representation that is able to identify
locations where categories are over- or underrepresented. From there, we
provide a new measure of exposure that discriminates between situations where
categories co-locate or repel one another. We then use this feature to provide
an unambiguous, parameter-free method to find meaningful breaks in the income
distribution, thus defining classes. Applied to the 2014 American Community
Survey, we find 3 emerging classes -- low, middle and higher income -- out of
the original 16 income categories. The higher-income households are
proportionally more present in larger cities, while lower-income households are
not, invalidating the idea of an increased social polarisation. Finally, using
the density -- and not the distance to a center which is meaningless in
polycentric cities -- we find that the richer class is overrepresented in high
density zones, especially for larger cities. This suggests that density is a
relevant factor for understanding the income structure of cities and might
explain some of the differences observed between US and European cities.
