In Arthur’s trace formula, a ubiquitous role is played by certain limiting expressions arising from piecewise smooth functions with respect to projections of the Coxeter fan ($(G,M)$-families). These include terms resulting from intertwining operators on the spectral side and volumes of polytopes on the geometric side. We introduce the combinatorial concept of a compatible family with respect to an arbitrary polyhedral fan and obtain new formulas for the corresponding limiting expressions in this general framework. Our formulas can be regarded as algebraic generalizations of certain volume formulas for convex polytopes. In a companion paper, the results are used to study the spectral side of the trace formula.