  We combine theoretical results from polytope domain meshing, generalized
barycentric coordinates, and finite element exterior calculus to construct
scalar- and vector-valued basis functions for conforming finite element methods
on generic convex polytope meshes in dimensions 2 and 3. Our construction
recovers well-known bases for the lowest order N\'ed\'elec, Raviart-Thomas, and
Brezzi-Douglas-Marini elements on simplicial meshes and generalizes the notion
of Whitney forms to non-simplicial convex polygons and polyhedra. We show that
our basis functions lie in the correct function space with regards to global
continuity and that they reproduce the requisite polynomial differential forms
described by finite element exterior calculus. We present a method to count the
number of basis functions required to ensure these two key properties.
