  We show that discrete $W_m$ lattices are bi-Hamiltonian, using geometric
realizations of discretizations of the Adler-Gel'fand-Dikii flows as local
evolutions of arc length-parametrized polygons in centro-affine space. We prove
the compatibility of two known Hamiltonian structure defined on the space of
geometric invariants by lifting them to a pair of pre-symplectic forms on the
space of arc length parametrized polygons. The simplicity of the expressions of
the pre-symplectic forms makes the proof of compatibility straightforward. We
also study their kernels and possible integrable systems associated to the
pair.
