  We explicate the combinatorial/geometric ingredients of Arthur's proof of the
convergence and polynomiality, in a truncation parameter, of his non-invariant
trace formula. Starting with a fan in a real, finite dimensional, vector space
and a collection of functions, one for each cone in the fan, we introduce a
combinatorial truncated function with respect to a polytope normal to the fan
and prove the analogues of Arthur's results on the convergence and
polynomiality of the integral of this truncated function over the vector space.
The convergence statements clarify the important role of certain combinatorial
subsets that appear in Arthur's work and provide a crucial partition that
amounts to a so-called nearest face partition. The polynomiality statements can
be thought of as far extensions of the Ehrhart polynomial. Our proof of
polynomiality relies on the Lawrence-Varchenko conical decomposition and
readily implies an extension of the well-known combinatorial lemma of
Langlands. The Khovanskii-Pukhlikov virtual polytopes are an important
ingredient here. Finally, we give some geometric interpretations of our
combinatorial truncation on toric varieties as a measure and a Lefschetz
number.
