3,465 research outputs found
Minimal Riesz Energy Point Configurations for Rectifiable d-Dimensional Manifolds
For a compact set A in Euclidean space we consider the asymptotic behavior of
optimal (and near optimal) N-point configurations that minimize the Riesz
s-energy (corresponding to the potential 1/t^s) over all N-point subsets of A,
where s>0. For a large class of manifolds A having finite, positive
d-dimensional Hausdorff measure, we show that such minimizing configurations
have asymptotic limit distribution (as N tends to infinity with s fixed) equal
to d-dimensional Hausdorff measure whenever s>d or s=d. In the latter case we
obtain an explicit formula for the dominant term in the minimum energy. Our
results are new even for the case of the d-dimensional sphere.Comment: paper: 29 pages and addendum: 4 page
Quasi-uniformity of Minimal Weighted Energy Points on Compact Metric Spaces
For a closed subset of a compact metric space possessing an
-regular measure with , we prove that whenever
, any sequence of weighted minimal Riesz -energy configurations
on (for `nice' weights) is
quasi-uniform in the sense that the ratios of its mesh norm to separation
distance remain bounded as grows large. Furthermore, if is an
-rectifiable compact subset of Euclidean space ( an integer)
with positive and finite -dimensional Hausdorff measure, it is possible
to generate such a quasi-uniform sequence of configurations that also has (as
) a prescribed positive continuous limit distribution with respect
to -dimensional Hausdorff measure. As a consequence of our energy
related results for the unweighted case, we deduce that if is a compact
manifold without boundary, then there exists a sequence of -point
best-packing configurations on whose mesh-separation ratios have limit
superior (as ) at most 2
Mesh ratios for best-packing and limits of minimal energy configurations
For -point best-packing configurations on a compact metric
space , we obtain estimates for the mesh-separation ratio
, which is the quotient of the covering radius of
relative to and the minimum pairwise distance between points in
. For best-packing configurations that arise as limits of
minimal Riesz -energy configurations as , we prove that
and this bound can be attained even for the sphere.
In the particular case when N=5 on with the Euclidean metric, we
prove our main result that among the infinitely many 5-point best-packing
configurations there is a unique configuration, namely a square-base pyramid
, that is the limit (as ) of 5-point -energy
minimizing configurations. Moreover,
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