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 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, .
Key words and phrases:Best-packing, mesh norm, separation distance, quasi-uniformity, Riesz energy, covering constant
2000 Mathematics Subject Classification:Primary: 31C20, 65N50, 57N16; Secondary: 52A40, 28A78
The research of these authors was supported, in part, by the U. S. National Science Foundation under grant DMS-1109266.
Let be a compact infinite metric space with metric and let denote a configuration of points in . We are chiefly concerned with two ‘quality’ measures of ; namely, the separation distance of defined by
and the mesh norm of with respect to defined by
This quantity is also known as the fill radius or covering radius of relative to . The optimal values of these quantities are also of interest and we consider, for , the -point best-packing distance on given by
and the -point mesh norm of given by
where denotes the cardinality of set . A configuration of points in is called a best-packing configuration for if .
In the theory of approximation and interpolation (for example, by splines or radial basis functions (RBFs)), the separation distance is often associated with some measure of ‘stability’ of the approximation, while the mesh norm arises in the error of the approximation. In this context, the mesh-separation ratio (or mesh ratio)
can be regarded as a ‘condition number’ for relative to . If is a sequence of -point configurations such that is uniformly bounded in , then the sequence is said to be quasi-uniform on . Quasi-uniform sequences of configurations are important for a number of methods involving RBF approximation and interpolation (see [9, 13, 16, 17]).
We remark that in some cases it is easy to obtain positive lower bounds for the mesh-separation ratio. For example, if is connected, then . Furthermore, letting
denote the closed ball in with center and radius , then for any -point configuration whenever and have the property that for any and any , the annulus is nonempty. The diameter of is defined by
The outline of the paper is as follows. In Section 2 we present two simple but basic results concerning the mesh-separation ratio for best-packing configurations on general sets. In Section 3, we obtain lower bounds for this ratio for any best-packing configuration on the sphere in and, in Section 4, we study the special case of minimal Riesz -energy 5-point configurations on and determine their limiting best-packing configuration as . Section 5 is devoted to a brief discussion of some special best-packing configurations on .
2. Mesh-separation ratio for general sets
The following simple result is of the same spirit as that of Proposition 2.1 of .
Let be a compact infinite metric space. Then, for each , there exists an -point best-packing configuration on such that . In particular, this holds for any best-packing configuration having the minimal number of pairs of points such that .
Let be a best-packing configuration on having the minimal number of unordered pairs of points such that . If , then select a point such that for , and choose a point from some pair such that . Let be the best-packing configuration obtained by replacing in by . Clearly, has fewer unordered pairs of points such that than . This contradiction proves Theorem 1. ∎
On the other hand, there exist examples of compact metric spaces for which
as we now show.
Example 1. Let be the standard Cantor set in [0,1] and let be the Euclidean metric. For each , the set is contained in the union of disjoint intervals of length with endpoints which belong to . For any configuration of points in , at least one of the intervals of length must contain at least two points from the configuration showing that . On the other hand, the configuration is a best-packing configuration since and has mesh norm . Thus (5) holds.
Best-packing configurations arise as limits of minimum energy configurations as we now describe. For a configuration of distinct points and , the Riesz -energy of is defined by
while the -point Riesz -energy of is defined by
An -point configuration is said to -energy minimizing if .
Proposition 2 ().
Let be an infinite compact metric space. For each fixed ,
Moreover, every cluster point as of -energy minimizing -point configurations on is an -point best-packing configuration on .
The following theorem concerning the mesh-separation ratio of best-packing configurations that arise as cluster points of -energy minimizing configurations generalizes, simplifies, and improves Theorem 7 of .
For a fixed , let be a cluster point as of a family of -point -energy minimizing configurations on a compact metric space . Then .
The upper bound for in this theorem can be attained even for the case when is a sphere and is the Euclidean metric. For on , equality follows from the uniqueness result for best-packing of Böröczky . For on , it follows from Theorem 7 in Section 4.
Let be fixed and, for , let be an -point -energy minimizing configuration on . Clearly, . This implies that there exists a point such that
If then , where , and is a point of such that , for all , which yields a contradiction. Hence,
3. Lower bounds for the mesh-separation ratio on the sphere
In this section we derive some lower bounds for the mesh-separation ratio of a best-packing -point configuration on the unit sphere with the Euclidean metric. Let and be the sphere packing and covering constants in , respectively:
Let denote a sequence of -point best-packing configurations on . Then
It is interesting to investigate the asymptotic behavior of the constant on the right-hand side of (9) as . The best known asymptotic upper bound for is the Kabatyanski-Levenshtein bound as and the best known lower bound for the covering constant is , where is a positive absolute constant (cf. [7, pages 40 and 247]). Thus the inequality (9) implies the following: if is large enough and , then the inequality
holds for an arbitrary best-packing configuration on . Further upper bounds for and lower bounds for can be found in  and . In particular, it is known that for the hexagonal lattice provides both , and . Hence
for an arbitrary best packing configuration on . However, by special arguments working only for we are able to improve this result to the following:
Let denote a sequence of -point best-packing configurations on . Then
It suffices to only consider sequences such that as . For a fixed , consider the Voronoi decomposition of generated by , with denoting the cell associated with ; that is,
Euler’s formula for convex polyhedra implies that there is a cell having at most 5 edges (each cell is a spherical polygon with edges consisting of arcs of great circles), see . Since
and , it follows by a projection argument that there is at least one interior angle from to consecutive vertices of with angle , and hence the distance from to some vertex of is at least
This yields (12).
4. Limit of minimal energy for 5 points on
It was observed in  from numerical experiments that 5-point minimum Riesz -energy configurations on with the Euclidean metric appear to depend on and to be of two general types: (i) the bipyramid (BP) consisting of 2 antipodal points and 3 equally spaced points on the associated equator, and (ii) the square-base pyramid (SBP) with one vertex at the north pole and 4 vertices of the same latitude depending on and forming a square (see Figure 1). A comparison of the -energy for the BP and the SBP configurations is given in Figure 2 and suggests (as in ) that BP is optimal for , while SBP is optimal for .
R. Schwartz  using a mathematically rigorous computer-aided solution proved (in a manuscript of 67 pages) that, for , BP is the unique minimizer of the Riesz -energy for and . (For the logarithmic energy, the optimality of BP is established in .) Currently there are no other values of for which a rigorous optimality proof is known. Regarding the stability of BP and SBP(), in Figure 3 we plot the minimum eigenvalue of the Hessian of their -energies. These graphs suggest that BP is not a local minimizing configuration for (also observed by H. Cohn), while SBP() is not a local minimizing configuration for .
According to Proposition 2, every cluster point of -energy minimizing configurations as is a best-packing configuration. However, as is known, there are infinitely many non-isometric 5-point best-packing configurations on (see e.g. ).
and all 5-point best-packing configurations on consist of two antipodal points (poles) and a triangle on the equator having all angles greater than or equal to .
It appears from Figure 2 that the unique (up to isometry) cluster point of 5-point -energy minimizing configurations is SBP(); that is, the square base pyramid with base on the equator. We shall next provide a rigorous proof that this is indeed the case.
Let be a cluster point of a family of -point -energy minimizing configurations on as . Then is isometric to
where , , and .
It is perhaps surprising that this configuration has the maximum number of common pairwise distances (eight) of length among all -point best-packings.
We start the proof with an upper estimate for the minimum 5-point -energy on .
For arbitrary , we define the following 5-point configuration on :
which, for a suitable choice of (depending on ), is a conjectured minimum energy configuration on for every large enough. The -energy of this configuration is given by
Letting now , we obtain that
We further need the following statement.
Let , and be fixed positive constants. Then
for every and .
It is not difficult to see that attains its minimum on at the point if and at the point
if . In the first case we have
In the second case, since
for all and . Combining the results in both cases, we obtain the assertion of the lemma. ∎
Proof of Theorem 7..
As we mentioned in Proposition 2 above, any cluster point of a family of -energy minimizing configurations as is a best-packing configuration. Thus, by Proposition 6, it is sufficient to show that no 5-point configuration consisting of two opposite poles and an acute triangle on the equator (which we call an acute configuration) could be such a cluster point. We will prove this by contradiction. For large, consider a minimal -energy configuration that is ‘close’ to a fixed acute configuration. We may assume that this minimal -energy configuration consists of three points
where as , that are close to the vertices of a fixed acute triangle on the equator, and two points and that are close to and , respectively. Denote by
Clearly, the total -energy .
Let us first estimate from below. Denote by the point , by the projection of to the plane , and by the length . Without lost of generality we may assume that lies in the triangle . Here we use the facts that as , and that is ‘close’ to a fixed acute triangle implying that lies inside the triangle . Denote by , , and the angles , , , respectively (see Figure 4).
we have, by the law of cosines and the fact that ,
The crucial observation is the fact that , for some that does not depend on . Now monotonicity and convexity of the function , , immediately imply
From the facts that , and as and the inequality , we get that
for some absolute constant . Then, by Lemma 9,
for some absolute constant . Similarly we obtain
and so again applying the convexity of we finally deduce that, for sufficiently large,
We can now obtain the dominant term in the asymptotic expansion for the minimal 5-point -energy.
By Lemma 8 it is enough to prove that
For a fixed consider a minimal -energy configuration . By Theorem 7 we may assume that both distances and have limit as . Observe that if the triangle is not acute, then
where is the midpoint of the circular arc (of length less than ) joining and and containing . A similar statement holds for triangle . Therefore we may assume that at least one of the triangles or is acute since otherwise the desired lower bound for the -energy follows. Without lost of generality, we assume that is acute. We adopt the same notation as in the proof of Theorem 7 and obtain a finer lower bound for and .
There are three possible cases to consider, depending on the location of the projection of onto the plane containing , , and : (i) is inside the sector ; (ii) is inside the sector ; and (iii) is inside the sector .
In both other cases (ii) and (iii) we get the same inequality. Letting denote the projection of onto the plane and setting , we similarly get
5. Special best-packing configurations on
By the proof of Theorem 1, we have for some best-packing configuration if and only if , which should be a very rare event, at least for . For and there exists a unique (up to isometry) best-packing configuration consisting of the regular icosahedron minus one of its vertices (see ). Hence,
Assuming (19), we are able to construct a best-packing configuration of points on the sphere with . It consists of -cell without certain 7 points which we describe below.
In the -cell each point has 12 other points at the closest distance , and each pair of points at this distance has exactly 5 other points having the same distance to both points of the pair. So we will remove two points , such that , and also 5 points , such that , , . Recall that the second largest distance between points of the -cell is 1. Thus,
Acknowledgements. The authors thank the Mathematisches Forschungsinstitut Oberwolfach for their hospitality during the preparation of this manuscript and for providing a stimulating atmosphere for research.
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