A very general question in Geometric Measure Theory is "how does the regularity of a measure affect the geometry of its support?" An asymptotically optimally doubling measure on Rn is one which infinitesimally behaves like m-dimensional Lebesgue measure. David, Kenig, and Toro studied such measures under a mild flatness assumption. In this talk, we discuss the geometry of the support of such measures without any flatness assumptions.