2011/04/14 by Douglas P. Hardin, Edward B. Saff, Hardin, D. P. +3
Mathematics · #28A78 (Secondary) #31C20 #57N16 (Primary) 52A40 #65N50 #FOS: Mathematics #FOS: Physical sciences #Graph theory and applications #Mathematical Approximation and Integration #Mathematical Physics (math-ph) #Numerical Analysis (math.NA) #Point processes and geometric inequalities
paper · pdf · doi:10.48550/arxiv.1104.2911
openalex publication_date 2011/04/14 · openalex created_date 2019/06/27 · openalex updated_date 2026/08/01
For a closed subset K of a compact metric space A possessing an α-regular measure μ with μ(K)>0, we prove that whenever s>α, any sequence of weighted minimal Riesz s-energy configurations ωN=\xi,N(s)\i=1N on K (for `nice' weights) is quasi-uniform in the sense that the ratios of its mesh norm to separation distance remain bounded as N grows large. Furthermore, if K is an α-rectifiable compact subset of Euclidean space (α an integer) with positive and finite α-dimensional Hausdorff measure, it is possible to generate such a quasi-uniform sequence of configurations that also has (as N→ ∞) a prescribed positive continuous limit distribution with respect to α-dimensional Hausdorff measure. As a consequence of our energy related results for the unweighted case, we deduce that if A is a compact C1 manifold without boundary, then there exists a sequence of N-point best-packing configurations on A whose mesh-separation ratios have limit superior (as N→ ∞) at most 2.