Abstract: We extend two known constructions that relate regular subdivisions to initial degenerations of projective toric varieties and Grassmannians. We associate a point configuration A with any homogeneous ideal I. We obtain upper and lower bounds on each initial ideal of I in terms of regular subdivisions of A. We also investigate when these bounds are exact, for example, the lower bound is exact for every initial ideal of the Plücker ideal I(2,n) with respect to points in the tropicalization. This talk is based on joint work with Dan Corey, Igor Makhlin, and Victoria Schleis.