On point energies, separation radius and mesh norm for s-extremal configurations on compact sets in Rn

We investigate bounds for point energies, separation radius, and mesh norm of certain arrangements of N points on sets A from a class A of d-dimensional compact sets embedded in R ′ , 1 ≤ d ≤ d′. We assume that these points interact through a Riesz potential V = | · |−s, where s > 0 and | · | is the Euclidean distance in R ′ . With δ∗ s (A,N) and ρ ∗ s(A,N) denoting, respectively, the separation radius and mesh norm of s-extremal configurations, which are defined to yield minimal discrete Riesz s-energy, we show, in particular, the following. (A) For the d-dimensional unit sphere S ⊂ R and s < d− 1, δ∗ s (S, N) ≥ cN−1/(s+1) and, moreover, δ∗ s (S , N) ≥ cN−1/(s+2) if s ≤ d− 2. The latter result is sharp in the case s = d− 2. In addition, point energies for s-extremal configurations are asymptotically equal. This observation relates to numerical experiments on observed scar defects in certain biological systems. (B) For A ∈ A and s > d, δ∗ s (A,N) ≥ cN−1/d and the mesh ratio ρs(A,N)/δ s (A,N) is uniformly bounded for a wide subclass of A. We also conclude that point energies for s-extremal configurations have the same order, as N →∞. AMS Classification : Primary 28A78, 52A40; Secondary 31C05, 31C20, 57N16.