  The Choquet integral w.r.t. a capacity can be seen in the finite case as a
parsimonious linear interpolator between vertices of $[0,1]^n$. We take this
basic fact as a starting point to define the Choquet integral in a very general
way, using the geometric realization of lattices and their natural
triangulation, as in the work of Koshevoy. A second aim of the paper is to
define a general mechanism for the bipolarization of ordered structures. Bisets
(or signed sets), as well as bisubmodular functions, bicapacities,
bicooperative games, as well as the Choquet integral defined for them can be
seen as particular instances of this scheme. Lastly, an application to
multicriteria aggregation with multiple reference levels illustrates all the
results presented in the paper.
