Optimal $N$-Point Configurations on the Sphere: “Magic” Numbers and Smale’s 7th Problem

Abstract

This paper inquires into the concavity of the map $N\mapsto v_s(N)$ from the integers $N\ge 2$ into the minimal average standardized Riesz pair-energies $v_s(N)$ of $N$-point configurations on the sphere $\mathbb {S}^2$ for various $s\in \mathbb {R}$. The standardized Riesz pair-energy of a pair of points on $\mathbb {S}^2$ a chordal distance $r$ apart is $V_s(r)= s^{-1}\left( r^{-s}-1 \right)$, $s \ne 0$, which becomes $V_0(r) = \ln \frac{1}{r}$ in the limit $s\rightarrow 0$. Averaging it over the $\left( \begin{array}{c} N\\ 2\end{array}\right)$ distinct pairs in a configuration and minimizing over all possible $N$-point configurations defines $v_s(N)$. It is known that $N\mapsto v_s(N)$ is strictly increasing for each $s\in \mathbb {R}$, and for $s<2$ also bounded above, thus “overall concave.” It is (easily) proved that $N\mapsto v_{-2}^{}(N)$ is even locally strictly concave, and that so is the map $2n\mapsto v_s(2n)$ for $s<-2$. By analyzing computer-experimental data of putatively minimal average Riesz pair-energies $v_s^x(N)$ for $s\in \{-1,0,1,2,3\}$ and $N\in \{2,\ldots ,200\}$, it is found that the map $N\mapsto {v}_{-1}^x(N)$ is locally strictly concave, while $N\mapsto {v}_s^x(N)$ is not always locally strictly concave for $s\in \{0,1,2,3\}$: concavity defects occur whenever $N\in {\mathcal {C}}^{x}_+(s)$ (an $s$-specific empirical set of integers). It is found that the empirical map $s\mapsto {\mathcal {C}}^{x}_+(s),\ s\in \{-2,-1,0,1,2,3\}$, is set-theoretically increasing; moreover, the percentage of odd numbers in ${\mathcal {C}}^{x}_+(s),\ s\in \{0,1,2,3\}$ is found to increase with $s$. The integers in ${\mathcal {C}}^{x}_+(0)$ are few and far between, forming a curious sequence of numbers, reminiscent of the “magic numbers” in nuclear physics. It is conjectured that these new “magic numbers” are associated with optimally symmetric optimal-log-energy $N$-point configurations on $\mathbb {S}^2$. A list of interesting open problems is extracted from the empirical findings, and some rigorous first steps toward their solutions are presented. It is emphasized how concavity can assist in the solution to Smale’s $7$th Problem, which asks for an efficient algorithm to find near-optimal $N$-point configurations on $\mathbb {S}^2$ and higher-dimensional spheres.

Appendices

Appendix 1: Optimal Riesz Energy Configurations on $\mathbb {S}^2$: A Brief Survey

The problem to determine $v_s (N)$ together with the minimizing configuration(s) $\omega _N^s$ has been solved completely for all $N\ge 2$ only at the distinguished value $s=-2$, by explicit calculation. Namely, $s=-2$ yields the energy law for the completely integrable Newtonian $N$-body problem with repulsive harmonic forces. Any $N$-point configuration satisfying $\sum _{i=1}^N {\varvec{{ q}} }_i = \mathbf {0}$ is a minimizing configuration of $\langle V_s\rangle (\omega _N)$, and only such are. The minimal energy reads

$$\begin{aligned} v_{-2}(N) = -\frac{1}{2}\frac{N+1}{N-1}. \end{aligned}$$

(109)

The (presumably) next-simplest parameter regime is $s<-2$. Here one is confronted with the possibly startling observation that for large $N$ the $N$-tuple Fekete points accumulate around two opposite points, and the localization sharpens as $N$ is getting larger; this is a consequence of Theorem 7 in [38]. In particular, it follows right away from Theorem 7 in [38] that for even $N$ the infimum $v_s (N),\, s<-2$, is achieved³⁴ if and only if half of the particles each are placed at two antipodal points,³⁵ yielding

$$\begin{aligned} v_s (N) = -\frac{1}{|s|}\,\frac{\left( 2^{|s|-1}-1 \right) N+1}{N-1},\qquad s< -2,\qquad N=2n, \end{aligned}$$

(110)

which converges to $v_{-2}(N)$ when taking the limit $s\uparrow -2$ of (110). When $N$ is odd the situation is already more tricky, and more interesting! For instance, for the smallest allowed odd $N=3$ it is suggestive to conjecture that the minimizing configuration consists of the corners of an equilateral triangle in an arbitrary equatorial plane; yet comparison with an antipodal “configuration” (arrangement) with two labeled points in the North and one in the South Pole reveals that the equilateral configuration yields a lower average standardized Riesz pair-energy only for $s_3 < s < -2$, where $s_3 \equiv \ln (4/9)/\ln (4/3)$, while for $s < s_3$ the antipodal arrangement yields the lower average standardized Riesz pair-energy; in this case one can easily show rigorously that the antipodal arrangement is in fact optimal: namely, the equilateral triangle and the antipodal arrangement are the only equilibrium arrangements of 3 labeled points. When comparing the average standardized Riesz pair-energy for antipodal and equilateral arrangements for other odd $N$, this changeover happens only if $N$ is a multiple of $3$. The critical $s_{3(2n-1)}$ tends monotonically to $-2$ as $N =3(2n-1)\rightarrow \infty$. Of course, this does not prove that either arrangement is optimal in the respective range of $s$. To the best of our knowledge, the optimal arrangement of odd-$N$ points as a function of $s<-2$ is far from being settled.

The concentration of the minimizing “$N$-point configuration” for $s< -2$ at a few distinct points indicates that the optimization problem is incorrectly posed in the set of proper $N$-point configurations. (The deeper reason is that the Riesz pair-energy ceases to be positive definite in the sense of Schoenberg [43] for $s < -2$.) Interestingly, the sum of distance problem for $s < -2$ plays a central role in the theory of Quasi-Monte Carlo integration schemes³⁶ for functions in smooth enough function spaces over $\mathbb {S}^2$; we refer the interested reader to [48–50] and papers cited therein.

When $s> -2$ the problem becomes drastically more complicated. One needs to distinguish the cases $-2<s<2$, $s=2$, $s>2$, and the limit ${s\rightarrow \infty }$.

The interval $-2<s<2$ is known as the potential-theoretical regime, since concepts and methods of potential theory can be applied to study both the discrete and the continuous (i.e. $N\rightarrow \infty$) optimization problems.³⁷ Within this regime the integer values $s=-1$, $s=0$, and $s=1$ are of particular interest. When $s=-1$ the minimal average standardized Riesz pair-energy problem is equivalent to the maximal average pairwise chordal distance problem; see [52–54]. In [49] it is shown that maximum-sum-of-distance configurations are ideal integration nodes for a certain optimal-order Quasi-Monte Carlo integration scheme on $\mathbb {S}^2$; we will come back to this in Appendix . The case “$s=0$,” i.e. the limit $s\rightarrow 0$, which yields the logarithmic pair-energy 2 (also known as the Coulomb energy for a pair of “two-dimensional unit point charges” on $\mathbb {S}^2$, respectively the Kirchhoff energy [55] of a pair of unit point vortices on $\mathbb {S}^2$), occurs in a stunning variety of problems (on $\mathbb {S}^2$ and other manifolds) in the sciences and mathematics; see, e.g. [10, 56–63]. Originally Smale’s 7th problem for the 21st century [19] was formulated for the logarithmic energy. Lastly, the value $s=1$ yields the Coulomb pair-energy of “three-dimensional unit point charges” associated with the so-called Thomson problem (see [5, 8, 9, 18, 45, 46, 64]).

Amongst the values $s\ge 2$, the borderline value $s=2$ is special in the sense that the finite-$N$ behavior is qualitatively different from both, the regime $-2<s<2$, and the regime $s>2$. Yet it can be understood by considering a certain limit process $s \rightarrow 2$; cf. [65] for the limit process $s\rightarrow d$ in analogous optimization problems formulated on $d$-dimensional manifolds. The Riesz pair interaction for $s=2$, in physics considered as correction term to Newton’s gravity [66], is also special in the sense that it yields a Newtonian $N$-body problem in $\mathbb {R}^3$ with additional isolating integrals of motion [67–69], besides those associated with Galilei invariance. Restricted to $\mathbb {R}$ the motion is even completely integrable for all $N$ [70, 71].

The large-$s$ behavior of $\langle V_s\rangle (\omega _N)$ ($N$ fixed) is intimately connected with the classical Tammes’s problem ([72]) or hard sphere (best-packing) problem (cf. [73]); that is, to find a configuration of $N$ points on the sphere with the minimal pairwise (chordal) distance between the points being as large as possible.³⁸ It is not too difficult to see (cf. Appendix ) that for any $N$-point configuration $\omega _N = \{ {\varvec{{ q}} }_1, \dots , {\varvec{{ q}} }_N \} \subset \mathbb {S}^2$ the following limit relation holds:

$$\begin{aligned} \lim _{s \rightarrow \infty } \left[ \langle V_s\rangle (\omega _N) +{\textstyle \frac{1}{s}}\right] ^{-1/s} = \min _{1 \le i < j \le N} \left| {\varvec{{ q}} }_i - {\varvec{{ q}} }_j \right| \equiv \varrho ( \omega _N ). \end{aligned}$$

(111)

Moreover, whenever a family of minimizing configurations $\omega _N^s$ converges to a limit configuration $\omega _N^\infty$ one has the relation (cf. Appendix )

$$\begin{aligned} \lim _{s \rightarrow \infty } \left[ v_s (N) +{\textstyle \frac{1}{s}}\right] ^{-1/s} = \varrho ( \omega _N^\infty ) \equiv \rho (N), \end{aligned}$$

(112)

where $\rho (N)$ is the best-packing (chordal) distance, which maximizes the least distance $\varrho (\omega _N)$ among all $N$-point configurations on $\mathbb {S}^2$, and $\omega _N^\infty$ is the best-packing configuration.³⁹ The best-packing distance $\rho (N)$ is only known for $N = 2,\ 3,\ \dots ,\ 12$ and $24$ (cf. [77–81]).⁴⁰

Another way of obtaining a nontrivial limit problem is to let $s\rightarrow \infty$ in $V_s(r)$, which gives

$$\begin{aligned} V_{\infty }(r) \equiv \lim _{s\rightarrow \infty } V_{s}(r) = {\left\{ \begin{array}{ll} \infty &{} \text {if } r < 1, \\ 0 &{} \text {if } r \ge 1. \end{array}\right. } \end{aligned}$$

(113)

In that case $v_\infty (N) = 0$ for $N \le N_*$, while $v_\infty (N) = \infty$ for $N > N_*$, viz. $N_*$ is the maximum number of non-overlapping calottes with spherical radius $\pi /6$ which can be placed on the unit sphere. By picking any ball, $B$, from a hexagonal close packing (hcp) of $\mathbb {R}^3$ with unit balls, then projecting its 12 nearest neighbors (unit balls) radially onto the surface of the central ball $B$, one sees that $N_*\ge 12$. And dividing the surface area of the unit sphere, $4\pi$, by the area of a calotte, $\pi (2-\sqrt{3})$, yields $\approx 14.92820323$, giving the upper bound $N_*< 15$. But how large is $N_*$, exactly?

Interestingly, the sharp value for $N_*$ is found by studying the related Tammes problem. For $2 \le N \le 12$ one has⁴¹ $\rho (N) > 1$. L. Fejes Tóth’s famous inequality ([79]),

$$\begin{aligned} \left[ \rho (N) \right] ^2 \le 4 - \Bigl [ \mathrm {cosec}\Big ( \frac{\pi }{6} \frac{N}{N-2} \Big ) \Bigr ]^2, \end{aligned}$$

where equality holds only for $N = 3$, $4$, $6$ and $12$, gives $\rho (N) < 1$ for $N \ge 14$. From [83] follows $\rho (13) < 1$. Hence, $N_* = 12$.

To our best knowledge, the following point sets are the only ones for which one can rigorously prove that they have minimal average standardized Riesz pair-energy for all $s > -2$. One can easily characterize the minimizing configuration explicitly only when $N = 2$ or $3$ (as the antipodal and equilateral configuration, respectively). The minimizing configuration has been characterized explicitly also for $N = 4$, $6$, and $12$ as the vertices of Platonic solids⁴² (tetrahedron, octahedron, and icosahedron), which are known to be universally optimal (see [84]); such configurations minimize the potential energy of completely monotonic pair-energy functions. The standardized Riesz pair-energies for $s > -2$ (including the logarithmic pair-energy at $s=0$) fall into this category. The listed configurations for $N = 2$, $3$, $4$, $6$, and $12$ exhaust the possibilities for universally optimal configurations on $\mathbb {S}^2$; cf. [84, 85].

The surprisingly difficult task of finding a proof of minimality can, perhaps, be best illustrated with the only partly resolved five point problem on $\mathbb {S}^2$. It is clear from [84, Prop. 14] that there is no universally optimal $5$-point configuration on $\mathbb {S}^2$. Indeed, computational optimization reveals that the minimal-energy arrangement of five labeled points on $\mathbb {S}^2$ changes many times as $s$ varies over the real line.⁴³ Thus, for $s \le -2.368335\ldots$ an antipodal arrangement with two labeled points in the South, and three in the North Pole (say) is the optimizer; at $s=-2.368335\ldots$ a crossover takes place, and for $-2.368335\ldots \le s \le -2$ the energy-minimizing arrangement of five labeled points is an isosceles triangle on a great circle, with one point in the North Pole and two labeled points each in the other two corners, with (numerically) optimized height. At $s=-2$ the isosceles arrangement bifurcates off of a continuous family of rectangular pyramids with height $h=5/4$, all of which have the same energy $-3/4$ at $s=-2$, and of which the isosceles arrangement is the degenerate limit. At $s=-2$ also another crossover happens, and for $-2\le s\le 15.048077392\dots$ the regular triangular bi-pyramid is the putative energy-minimizing configuration. At $s = 15.048077392\dots$, yet another crossover happens, at which the triangular bi-pyramid and a square pyramid with height $h\approx 1.1385$ have the same average (standardized) Riesz pair-energy. A square pyramid with (numerically) optimized height⁴⁴ as function of $s$ appears to have lower (standardized) Riesz pair-energy for $s\in [15.04807\ldots ,\infty )$. Lastly, it is well-known that the triangular bi-pyramid and the square pyramid with height $1$ both are particular best-packing configurations, with $\varrho (\omega _5^\infty )=\sqrt{2}$, so that “at $s=\infty$” an “asymptotic crossover, or degenerate bifurcation,” happens.

How much of this has been proved rigorously? By traditional methods (see [86]) it can be shown that the triangular bi-pyramid consisting of two antipodal points at, say, the North and the South Pole, and three equally spaced points on the Equator, is the unique (up to orthogonal transformation) minimizer of the logarithmic average pair-energy. The proof that the same configuration maximizes the sum of distances (that is: assumes $v_{-1}(5)$) is computer-aided, exploiting interval methods and related techniques (see [87]). In [88] a computer-aided approach is proposed to show optimality of the triangular bi-pyramid for $s = 1$ and $s = 2$. The optimality of both the triangular bi-pyramid and the family of rectangular pyramids with height $5/4$ at $s = -2$ can be shown with elementary techniques. The rest of the $s$-parameter regime still awaits its rigorous treatment.⁴⁵

Numerical results in [36], carried out with varying $s\in [1,400]$ for $N\in \{2,\ldots ,16\}$ fixed, suggest that also for other values of $N\not \in \{2,3,4,6,12\}$ the minimizing configuration $\omega _N^s$ may generally change as $s$ passes through critical values, and their number seem to depend on $N$. In particular, for $N=7$ there seem to be three(!) critical $s$-values in $[1,6]$ at which the minimizing configuration changes, and presumably a few more when $s<1$, cf. [37] for $s=-1$. The general dependence on $s$ of the optimal $N$-point configurations $\omega _N^s$ is one of the intriguing features of this minimization problem. All the same, it makes it plain why the rigorous determination of the optimizers is a highly nontrivial task even for moderate $N$-values other than the special ones $2,3,4,6,12$, becoming hopelessly complicated when $N$ increases.

Yet, the large-$N$ asymptotics of the minimal average standardized Riesz pair-energy $v_s (N)$ can be determined without seeking the exact Fekete points, see [6]. In particular, $\lim _{N\rightarrow \infty }v_s (N)$ is for all $s$ determined by the variational principle⁴⁶ (see [26, 38, 89, 90])

$$\begin{aligned} \lim _{N\rightarrow \infty }v_s (N) = \inf _{\mu \in \mathfrak {P}(\mathbb {S}^2)} \int \int _{\mathbb {S}^2\times \mathbb {S}^2} V_s(|{\varvec{{ p}} }-{\varvec{{ q}} }|)\mu (d{\varvec{{ p}} })\mu (d{\varvec{{ q}} }); \end{aligned}$$

(114)

here, $\mathfrak {P}(\mathbb {S}^2)$ is the set of all Borel probability measures supported on $\mathbb {S}^2$. For $s\le -2$ the minimizer is not unique, but all minimizers are known; in particular, for $s<-2$ the minimizer, after factoring out $SO(3)$, is a symmetric measure which is concentrated on two antipodal points, see [38]. From classical potential theory (cf. [26] for $s\in [0,2)$ and [38] for $s\in (-2,0)$)⁴⁷ it is well-known that the uniform normalized (Lebesgue) surface area measure on $\mathbb {S}^2$, denoted by $\sigma$, uniquely minimizes the right-hand side in (114) for $-2 < s < 2$. For $s\ge 2$ the l.h.s. and r.h.s. of (114) are both $\infty$; in this case the rate of divergence of $v_s (N)$ can be determined. It “suffices” to know that for large $N$ the Voronoi cells around the charges are mostly hexagons of a certain size; see [91] for an enlightening discussion.⁴⁸ The picture one should have in mind, when $N$ is large, is a vast sea of hexagonal Voronoi cells around most of the points. Thus, the dual structure of the Voronoi cell decomposition, the Delaunay triangulation, is a network of mostly six-fold coordinated sites. The reason for the qualification “mostly” lies in the topology of the sphere, which gives rise to geometric frustration (see [92] for a thorough exposition of this notion). Certain points “pick up” a topological charge that measures the departure from the ideal coordination number, $6$, of the planar triangular lattice. The celebrated Euler theorem of topology yields that the total topological charge on $\mathbb {S}^2$ is always 12. This accounts, for example, for the appearance of 12 (isolated) pentagons in the common soccer ball design. For large $N$ one observes “scars” emerging from these isolated centers that attract pentagon-heptagon pairs (having total topological charge $0$). Scars and other topology-induced defects of the hexagonal lattice become important when pushing the asymptotic analysis to higher order, and are not well understood. For instance, to the best of our knowledge it is an unresolved question if there are $n$-gon Voronoi cells with $n \ge 8$ in a minimizing configuration. See [93] for an approach using elastic continuum formalism.

The truly hard regime is the vast intermediate range of $N$ which are generically too large to allow for an explicit determination of the minimizing configuration, but not large enough for the asymptotic formulas to yield sufficiently accurate results.

This concludes our brief survey of this fascinating field. Further information can be found in the survey articles [5, 41],[91], and on the websites [18] and [94]. See also the delightful article [95] where, based on numerical evidence, the first dozen minimizers are discussed mostly for Thomson’s problem ($s=1$).

Appendix 2

In this appendix we supply the proofs of relations (111) and (112) which control the limit $s\rightarrow \infty$.

1.1 Proof of Relation (111)

Suppose $\omega _N = \{ {\varvec{{ q}} }_1, \dots , {\varvec{{ q}} }_N \} \subset \mathbb {S}^2$ is a fixed $N$-point set, with separation distance $\varrho ( \omega _N ) = \min _{1 \le i < j \le N} | {\varvec{{ q}} }_i - {\varvec{{ q}} }_j |$. Then using that the function $f(x) \equiv x^{-1/s}$ is strictly decreasing for $s > 0$, we find that

$$\begin{aligned} \left[ \langle V_s\rangle (\omega _N) +{\frac{1}{s}}\right] ^{-1/s}&= \left[ \frac{1}{s} \frac{2}{N(N-1)} \mathop {\sum \sum }_{1 \le i < j \le N} \frac{1}{\left| {\varvec{{ q}} }_i - {\varvec{{ q}} }_j \right| ^{s}} \right] ^{-1/s} \\&= \varrho ( \omega _N ) \left[ \frac{1}{s} \frac{2}{N(N-1)}\mathop {\sum \sum }_{1 \le i < j \le N} \left( \frac{\varrho (\omega _N)}{\left| {\varvec{{ q}} }_i - {\varvec{{ q}} }_j\right| }\right) ^{s}\right] ^{-1/s}\\&\ge \varrho ( \omega _N ) \left( \frac{1}{s} \right) ^{-1/s} \\&\ge \varrho ( \omega _N ). \end{aligned}$$

On the other hand, retaining only one of the least distance pairs in the double sum yields

$$\begin{aligned} \left[ \langle V_s\rangle (\omega _N) +{\frac{1}{s}}\right] ^{-1/s}&\le \left( \frac{1}{s} \frac{2}{N(N-2)} \frac{1}{\varrho (\omega _N)^s}\right) ^{-1/s}\\&= \varrho ( \omega _N ) \left( \frac{1}{s} \right) ^{-1/s} \left( \frac{2}{N(N-2)} \right) ^{-1/s} \\&\rightarrow \varrho ( \omega _N )\qquad \text{ as }\quad s\rightarrow \infty . \end{aligned}$$

This completes the proof of (111).

1.2 Proof of Relation (112)

Let $\omega _N^s = \{ {\varvec{{ q}} }_1^s, \dots , {\varvec{{ q}} }_N^s \} \subset \mathbb {S}^2$ denote a minimizing $N$-point set, and suppose $\omega _N^\infty$ is a best-packing configuration with $\varrho ( \omega _N^\infty ) = \rho (N)$.

Then, first of all,

$$\begin{aligned} \langle V_s \rangle (\omega _N^s ) \le \langle V_s \rangle ( \omega _N^\infty ); \end{aligned}$$

but $\langle V_s \rangle (\omega _N^s ) = v_s (N)$, and so, by (111), we have

$$\begin{aligned} \liminf _{s \rightarrow \infty } \left[ v_s (N) +{\textstyle \frac{1}{s}}\right] ^{-1/s} \ge \varrho ( \omega _N^\infty ) \equiv \rho (N). \end{aligned}$$

(115)

On the other hand, using that $| {\varvec{{ q}} }_i^s - {\varvec{{ q}} }_j^s | = \varrho ( \omega _N^s )$ for at least one pair $(i,j)$, and furthermore that $\varrho ( \omega _N^s ) \le \rho (N)$, we obtain

$$\begin{aligned} \langle V_s \rangle (\omega _N^s ) + {\frac{1}{s}}&= {\frac{1}{s} \frac{2}{N(N-1)}} \mathop {\sum \sum }_{1 \le i < j \le N} \left| {\varvec{{ q}} }_i^s -{\varvec{{ q}} }_j^s\right| ^{-s} \\&= \left[ \rho (N) \right] ^{-s} \frac{1}{s} \frac{2}{N(N-1)} \mathop {\sum \sum }_{1 \le i < j \le N} \left[ \frac{\rho (N)}{\left| {\varvec{{ q}} }_i^s - {\varvec{{ q}} }_j^s \right| }\right] ^{s} \\&\ge \left[ \rho (N) \right] ^{-s} \frac{1}{s} \frac{2}{N(N-1)} > 0. \end{aligned}$$

Hence,

$$\begin{aligned} \left[ \langle V_s \rangle ( \omega _N^s ) + {\frac{1}{s}}\right] ^{-1/s}&\le \rho (N) s^{1/s} \left( N(N-1)/2 \right) ^{1/s} \\&\rightarrow \rho (N)\qquad \text{ as }\quad s\rightarrow \infty ; \end{aligned}$$

but again, $\langle V_s \rangle (\omega _N^s ) = v_s (N)$, and so we get

$$\begin{aligned} \limsup _{s \rightarrow \infty } \left[ v_s (N) +{\textstyle \frac{1}{s}}\right] ^{-1/s} \le \rho (N). \end{aligned}$$

(116)

By (115) and (116)

$$\begin{aligned} \lim _{s \rightarrow \infty } \left[ v_s (N) +{\textstyle \frac{1}{s}}\right] ^{-1/s} = \rho (N). \end{aligned}$$

This completes the proof of (112).

Appendix 3

We now prove the strict monotonic increase of $s\mapsto v_s (N)$.

1.1 Proof of Relation (14)

We begin with the observation that the map $s\mapsto V_s (r)$ is monotone increasing for all $r\in (0,2]$, in fact strictly so except when $r = 1$. To see this, take the first partial derivative of $V_s (r)$ w.r.t. $s$ to get

$$\begin{aligned} \partial _s V_s (r) = -s^{-2}\left( r^{-s}-1\right) -s^{-1} r^{-s} \ln r. \end{aligned}$$

(117)

Clearly, r.h.s.(117) $=0$ if $r=1$. We now show that r.h.s.(117) $>0$ if $r\ne 1$.

To this end, now take the first partial derivative of $\partial _s V_s (r)$ w.r.t. $r$ to get

$$\begin{aligned} \partial _{r} \partial _{s} V_s (r) = r^{-s-1} \ln r. \end{aligned}$$

(118)

Clearly, r.h.s.(118) $=0$ iff $r=1$; thus, $r\mapsto \partial _s V_s (r)$ has a critical point at $r=1$, and only this one. Finally, take the second partial derivative of $\partial _s V_s (r)$ w.r.t. $r$ to get

$$\begin{aligned} \partial _{r}^{2} \partial _{s} V_s (r) = r^{-s-2}\left( 1- (1+s) \ln r \right) . \end{aligned}$$

(119)

Evaluating r.h.s.(119) at $r=1$ yields $\partial _{r}^{2} \partial _{s} V_s (1)=1$; thus, for each $s$ the map $r\mapsto \partial _s V_s (r)$ has a non-degenerate minimum at $r=1$ with value $0$, and no other critical point. Therefore, $\partial _s V_s (r)\ge 0$, with “$=0$” iff $r=1$.

With the help of this calculus result we now conclude that whenever $s>t$, then

$$\begin{aligned} v_s (N) = \langle V_s\rangle (\omega _N^s) > \langle V_t\rangle (\omega _N^s) \ge \langle V_t\rangle (\omega _N^t) = v_t (N); \end{aligned}$$

(120)

here, the strict inequality holds because there is no optimizing $N$-point configuration on $\mathbb {S}^2$ with all pairs having distance 1.

This completes the proof.

Appendix 4: Spherical Digital Nets

Maximal sum-of-distance points (i.e. optimal configurations $\omega _N^s$ for $s = -1$) provide optimal integration nodes for equal-weight numerical integration rules on the sphere; see [49] and [50, 96]. In general, such configurations are obtained by solving a highly non-linear optimization problem which makes them impractical for large number of points.

Digital nets and sequences introduced in [97] are efficiently computable so-called low-discrepancy point systems in the unit square that define effective Quasi-Monte Carlo rules for integrating functions on the unit square

$$\begin{aligned} \frac{1}{N} \sum _{j=1}^N f( {\varvec{{ x}} }_j ) \approx \int \int _{[0,1]^2} f( {\varvec{{ x}} }) \, d {\varvec{{ x}} }. \end{aligned}$$

Informally speaking, the points of a digital net in $[0,1]^2$ are distributed in such a way that a large number of elementary rectangles contain precisely the fraction of all points that corresponds to their area; cf. [98] and Fig. 15.

Fig. 15

$2048$ Sobol’ points in the unit square generated using a method by [99]

This distribution property remains unchanged when lifting a digital net and elementary rectangles (now called spherical rectangles) to $\mathbb {S}^2$ using the area-preserving Lambert transformation of the map makers. These spherical digital nets are studied in [100] and [101]. Of particular interest is that numerically they are comparable with maximal sum-of-distance points. It is conjectured⁴⁹ that $\langle V_{-1} \rangle ( \omega _N^{\mathrm {sphDN}} )$ will approach the same limit as $v_{-1}(N)$ with the same rate of convergence as $N\rightarrow \infty$.

We tested for local concavity a sequence of $N$-point spherical digital nets formed by the first $N$ points of a Sobol’ sequence lifted to the sphere. For the implementation of the Sobol’ points we used [99]; cf. Fig. 15. The graph of $N \mapsto \langle V_{-1} \rangle ( \omega _N^{\mathrm {sphDN}} )$ (Fig. 16) shows an overall concave shape.

Fig. 16

$\langle V_{-1} \rangle ( \omega _N^{\mathrm {sphDN}} )$ based on the $2048$ Sobol’ points given in Fig. 15

Yet, some irregularities are clearly discernible in Fig. 16; in fact, the discrete second derivative of $N\mapsto \langle V_{-1} \rangle ( \omega _N^{\mathrm {sphDN}})$ reveals that $N\mapsto \langle V_{-1} \rangle ( \omega _N^{\mathrm {sphDN}})$ for this sequence of spherical digital nets is locally highly non-concave; see Fig. 17.

Fig. 17

Discrete second derivative of $\langle V_{-1} \rangle ( \omega _N^{\mathrm {sphDN}} )$ based on the $2048$ Sobol’ points given in Fig. 15. The discrete points are joint by lines to guide the eye