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

• Authors: S. B. Damelin;V. Maymeskul
• Venue: Journal of Complexity
• Year: 2005

We investigate bounds for point energies, separation radius, and mesh norm of certain arrangements of N points on sets A from a class Ad of d-dimensional compact sets embedded in Rd', 1 ≤ d ≤ d'. We assume that these points interact through a Riesz potential V = |ċ|-s, where s 0 and |ċ| is the Euclidean distance in Rd'. With δs*(A, N) and ρs*(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 Sd ⊂ Rd + 1 and s d - 1, δs*(Sd, N) ≥ cN-1/(s+1) and, moreover, δs*(Sd, 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 detects in certain biological systems.(B) For A ∈ Ad and s d, δs*(A, N) ≥ cN-1/d and the mesh ratio ρs*(A, N)/δs*(A, N) is uniformly bounded for a wide subclass of Ad. We also conclude that point energies for s-extremal configurations have the same order, as N → ∞