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

We investigate bounds for point energies, separation radius, and mesh norm of certain arrangements of N points on sets A from a class of d-dimensional compact sets embedded in R^d′, 1dd^′. We assume that these points interact through a Riesz potential V=|·|^-s, where s>0 and |·| is the Euclidean distance in . With and 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 and s<d-1, and, moreover, if sd-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 and s>d, and the mesh ratio is uniformly bounded for a wide subclass of . We also conclude that point energies for s-extremal configurations have the same order, as N→∞.