Sergiy V. Borodachov

Asymptotics of best-packing on rectifiable sets

We investigate the asymptotic behavior, as N grows, of the largest minimal pairwise distance of N points restricted to an arbitrary compact rec- tifiable set embedded in Euclidean space, and we find the limit distribution of such optimal configurations. For this purpose, we compare best-packing configurations with minimal Riesz s-energy configurations and determine the s-th root asymptotic behavior (as s ! 1) of the minimal energy constants. We show that the upper and the lower dimension of a set defined through the Riesz energy or best-packing coincides with the upper and lower Minkowski dimension, respectively. For certain sets in R d of integer Hausdor dimension, we show that the limiting behavior of the best-packing distance as well as the minimal s-energy for large s is dierent for dierent subsequences of the cardinalities of the configurations.

Low Complexity Methods For Discretizing Manifolds Via Riesz Energy Minimization

Let

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A

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A be a compact

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d

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d-rectifiable set embedded in Euclidean space