Research Article Symmetry Theorems and Uniform Rectifiability

Denote points in Euclidean n-space, Rn, by x = (x1, . . . ,xn) and let E and ∂E denote the closure and boundary of E ⊆Rn, respectively. Let 〈x, y〉 denote the standard inner product in Rn, |x| = 〈x,x〉1/2, and set B(x,r)= {y ∈Rn : |y− x| < r} whenever x ∈Rn, r > 0. Define k-dimensional Hausdorffmeasure, 1≤ k ≤ n, in Rn as follows: for fixed δ > 0 and E ⊆Rn, let L(δ)= {B(xi,ri)} be such that E ⊆ ⋃ B(xi,ri) and 0 < ri < δ, i= 1,2, . . . . Set