Addendum to Minimal Riesz Energy Point Configurations for Rectifiable D-dimensional Manifolds *

In the proof of Proposition 8.3 the assertion was made that each ψ i (K i) was contained in some open (relative to A) set G i such that (83) holds. The case s = d follows from the hypothesis that A is contained in a d-dimensional C 1 manifold since in this case the sets K i may be chosen to be almost clopen relative to A. The case s > d is more delicate and requires an argument similar to that used in the proof of Theorem 2.1 given in Section 7. In this addendum we provide a sketch of these arguments for the interested reader. In this note we let B d (a, r) denote the open ball in R d with center a and radius r, ¯ B d (a, r) the closure of B d (a, r), and C d (a, r) the boundary of B d (a, r). Let M ⊂ R d ′ be a C 1 manifold (without boundary), i.e. for each point p ∈ M there is some open set (relative to M) V ⊂ M containing p, some open ball B d (q, ρ), ρ > 0, and a diffeomorphic mapping φ p from B d (q, ρ) onto V (i.e. φ p is a homeomorphism of B d (q, ρ) onto V and φ p is continuously differentiable on B d (q, ρ)). By composing φ p with an appropriate affine mapping on R d we may assume without loss of generality that φ p (0) = p and that φ ′ p (0) T φ ′ p (0) = I so that φ ′ p (0) is an isometric mapping from R d onto its range in R d ′. For x, y ∈ B d (0, ρ), let [x, y] denote the directed line segment from x to y. Then φ p (y) − φ p (x) = [x,y] φ ′ p (z)dz = φ ′ p (0)(y − x) + [x,y] (φ ′ p (z(t)) − φ ′ p (0))dz(t). Minimal Riesz energy point configurations for rectifiable d-dimensional manifolds, Advances in Math., 1 Let ǫ > 0. From the continuity of φ ′ p and (1) it follows that there is some 0 < ν p < ρ such that φ p is bi-Lipschitz on B d (0, ν p) with constant (1 + ǫ). Since …