  A graphical design is a subset of graph vertices such that the weighted
averages of certain graph eigenvectors over the design agree with their global
averages. We use Gale duality to show that positively weighted graphical
designs in regular graphs are in bijection with the faces of a generalized
eigenpolytope of the graph. This connection can be used to organize, compute
and optimize designs. We illustrate the power of this tool on three families of
Cayley graphs -- cocktail party graphs, cycles, and graphs of hypercubes -- by
computing or bounding the smallest designs that average all but the last
eigenspace in frequency order.
