  We study the closed convex hull of various collections of Hilbert functions.
Working over a standard graded polynomial ring with modules that are generated
in degree zero, we describe the supporting hyperplanes and extreme rays for the
cones generated by the Hilbert functions of all modules, all modules with
bounded a-invariant, and all modules with bounded Castelnuovo-Mumford
regularity. The first of these cones is infinite-dimensional and simplicial,
the second is finite-dimensional but neither simplicial nor polyhedral, and the
third is finite-dimensional and simplicial.
