Nonflat points of Asymptotically Optimally Doubling Measures

We consider the question of how the doubling characteristic of a measure determines the regularity of its support. The question was first introduced by Kenig and Toro in their paper Free boundary regularity for Harmonic measure and Poisson kernels. The question was further considered by David, Kenig, and Toro for codimension-1 and later by Preiss, Tolsa, and Toro in higher codimension. However, their studies rely on a crucial assumption of flatness, and leave open the geometry of the support of such measures in a neighborhood about a non-flat point of the support. We here investigate the nonflat points.