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

Authors:
S. B. Damelin;V. Maymeskul
Affiliations:
Department of Mathematical Sciences, Georgia Southern University, Statesboro, GA 30460, USA;Department of Mathematical Sciences, Georgia Southern University, Statesboro, GA 30460, USA
Venue:
Journal of Complexity
Year:
2005

Citing 3
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Abstract

We investigate bounds for point energies, separation radius, and mesh norm of certain arrangements of N points on sets A from a class A^d of d-dimensional compact sets embedded in R^d^^^', 1=0 and |.| is the Euclidean distance in R^d^^^'. With @d"s^*(A,N) and @r"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^d@?R^d^+^1 and s=cN^-^1^/^(^s^+^1^) and, moreover, @d"s^*(S^d,N)=cN^-^1^/^(^s^+^2^) if s=d, @d"s^*(A,N)=cN^-^1^/^d and the mesh ratio @r"s^*(A,N)/@d"s^*(A,N) is uniformly bounded for a wide subclass of A^d. We also conclude that point energies for s-extremal configurations have the same order, as N-~.