Minimum Separation of the Minimal Energy Points on Spheres in Euclidean Spaces

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1 Minimum Separation of the Minimal Energy Points on Spheres in Euclidean Spaces A. B. J. Kuijlaars, E. B. Saff, and X. Sun November 30, 004 Abstract Let S d denote the unit sphere in the Euclidean space R d+1 (d 1). Let N be a natural number (N ), and let ω N := {x 1,..., x N } be a collection of N distinct points on S d on which the minimal Riesz s energy is attained. In this paper, we show that the points x 1,..., x N are well-separated for the cases d 1 s < d. 1 Introduction. Let S d denote the unit sphere in the Euclidean space R d+1 (d 1). Let N be a natural number (N ), and let ω N := {x 1,..., x N } be a collection of N distinct points on S d. The Riesz s energy (s 0) associated with ω N, E s (ω N ), is defined by i j E s (ω N ) := i j 1, if s > 0, x i x j s 1, if s = 0. log x i x j Here denotes the Euclidean norm. minimal s energy over S d defined by We use E s (S d, N) to denote the N point E s (S d, N) := min E s(ω N ), (1.1) ω N S d The first author is supported in part by FWO-Flanders projects G and G , by K.U.Leuven research grant OT/04/4, by INTAS Research Network NeCCA , and by NATO Collaborative Linkage Grant PST.CLG

2 where the infimum is taken over all N point subsets of S d. If ω N S d such that E s (ω N ) = E s (S d, N), then ω N is called a minimal s energy configuration, and the points in ω N are called minimal s energy points, or simply minimal energy points if the linkage to the parameter s is well-understood in a certain context. It is obvious that minimal s energy configurations exist. Also, if ω N is a minimal s energy configuration, and if ρ is a metric space isometry from S d to S d, then the image of ω N under ρ, ρ(ω N ), is also a minimal s energy configuration. Minimal (d 1) energy points are often referred to as Fekete points ; see [5]. The determination of minimal s energy configurations and the corresponding minimal s energy on S d and other manifolds is an important problem that has applications in many subjects including physics, chemistry, computer science, and mathematics. For more appreciation and understanding of this problem and its applications, we refer readers to the expository papers by Hardin and Saff [11], and by Saff and Kuijlaars [18]. The papers [6], [10], [1], and [16] and the references therein also contain valuable pertinent information. Full characterization of the distribution of minimal s energy points in the cases d turns out to be rather elusive. It is, however, generally expected that these points are well-separated in the sense that there exists a positive constant C d,s, depending only on d and s, such that min i j x i x j C d,s N 1/d. (1.) Dahlberg [5] proved that Fekete points are well-separated. (Dahlberg [5] actually established the well-separatedness of Fekete points on every compact C 1,α surface in R d+1 ). Kuijlaars and Saff [1] proved that minimal s energy points are well-separated for the cases s > d. There have been a series of more quantitative results regarding the case d =, s = 0, which corresponds to the minimal logarithmic energy on S. Rakhmanov, Saff, and Zhou [17] first showed that 3/5 is a lower bound for the constant C,0 in Inequality (1.). Dubickas [8] refined Rakhmanov, Saff, and Zhou s method, and showed that 7/4 is a lower bound for the constant C,0 in Inequality (1.). Using potential theory and stereographical projection technics, Dragnev [7] established an appealing lower bound for the minimum separation of the minimal logarithmic energy points on S to be (N 1) 1/. In this paper, we show that the minimal s energy points are well-separated for the cases d 1 s < d. Note that the case s = d 1 is already covered by the aforementioned Dahlberg s result. While the parameter s is restricted in the range d 1 s < d, the outcome s = 0 can only occur when d = 1, putting the problem on the unit circle

3 3 S 1. The configurations of minimal energy points on S 1 is well-understood; see [14]. Therefore in this paper, we can use the tacit assumption that s > 0. Our proof entails comparing the Riesz s potentials on the slightly larger sphere of radius 1 + N 1/d of two probability measures: the rotational invariant probability measure on S d, and the normalized counting measure on a minimal s energy configuration ω N. The Riesz s potential of the rotational invariant probability measure on S d can be expressed in close form in terms of the Gauss hypergeometric functions F 1 (a, b; c; z). The crux of our argument is an application of the principle of domination for α superharmonic functions; see [13]. This paper is organized as follows. In Section, we introduce the necessary notations and terminologies. Also in Section, we list a few formulas pertaining to the hypergeometric functions that we will use in our proofs. In Section 3, we state and prove our main result. Notation and terminology. Given a minimal s energy configuration ω N, we use v N counting measure on ω N, i.e., to denote the normalized v N := N 1 where δ xj denotes the unit point mass at x j. The rotational invariant probability measure on S d is denoted by µ. For a σ finite positive Borel measure λ supported on a compact subset K of R d+1, we define its Riesz s potential U s λ (s > 0) by U s λ(x) = x y s dλ(y). K Note that U s λ may take the extended value on some subsets of R d+1. In this paper, K is either S d or ω N. In the next section, we will show that the Riesz s potential of µ can be expressed in close form in terms of the Gauss hypergeometric functions F 1 (a, b; c; z) defined by F 1 (a, b; c; z) := N j=1 n=0 δ xj, where (a) n are the Pochhammer symbol defined by (a) n (b) n (c) n z n n!, (a) n = 1, if n = 0, (a) n = a(a + 1) (a + n 1), if n 1.

4 4 There are many good sources where one can find reliable information on the hypergeometric functions F 1 (a, b; c; z). We recommend Abramowitz et al [1], and Andrews et al []. We will be using a few basic formulas pertaining to the hypergeometric functions F 1 (a, b; c; z), which can all be found in [1]. We quote them here for readers easy reference. Under the conditions Rc > Rb > 0, the following formula holds true: 1 0 ( 1 zu) a u b 1 (1 u) c b 1 du = Γ(b)Γ(c b) Γ(c) F 1 (a, b; c; z). (.1) Formula.1 is often called Euler s integral representation for the hypergeometric function F 1. If R(c a b) < 0, then If R(c a b) > 0, then F 1 (a, b; c; z) Γ(c)Γ(a + b c) lim =. (.) z 1 (1 z) c a b Γ(a)Γ(b) F 1 (a, b; c; 1) = Γ(c)Γ(c a b) Γ(c a)γ(c b). (.3) The following derivative formula can be easily proved by term-by-term differentiation in the right domain: d dz F 1 (a, b; c; z) = ab c F 1 (a + 1, b + 1; c + 1; z). (.4) 3 Main Result and Proofs. On various occasions, we use C d,s to denote some unspecified positive constants, depending only on d and s. The exact values of C d,s may be different from proof to proof. In the same proof, however, we feel that it is imperative to use different notations for different constants. We will then use C d,s, and C d,s if necessary. Although in the current paper, we do not strive to estimate these constants, close forms of them can be obtained with some devoted calculations. We will be using the notations, and i j. The former denotes a double sum, excluding only those with i = j. The latter denotes a single sum in which i is fixed, and the summation is done only on j. We will emphasize this more when the latter is being used. The following lemma is well-known. For example, it can be found in [1].

5 5 Lemma 1. For d 1 s < d, there exists a positive constant C d,s independent of N, such that x i x j s > γ d,s N C d,s N 1+s/d, (3.1) where γ d,s := S d S d x y s dµ(x)dµ(y) = Γ((d + 1)/)Γ(d s) Γ((d s + 1)/)Γ(d s/). By inspecting the pertinent proof in [1], we find that a quantitative estimate of the constant C d,s in the above lemma can be made available if so desired. The behavior of this constant has received attentions in the literature, and in some special cases, conjectures have been made regarding this constant; see [3], [1], [17], and references therein. Lemma. For 0 < s < d, there exists a positive constant C d,s independent of N, such that U s v N (x) γ d,s C d,s N 1+s/d, x = 1. (3.) Proof: Since {x 1,..., x N } is a collection of N points on S d where the discrete s energy is attained, we have for each fixed i, 1 i N, x x j s x i x j s, x S d. Summing over i, and using Lemma 1, we get (N 1) N x x j s x i x j s γ d,s N C d,s N 1+s/d. (3.3) i j j=1 Dividing N(N 1) to get the desired the estimate. Lemma 3. For each fixed s > 0, the potential U s µ of the measure µ is a radial function, and has the explicit expression in terms of the hypergeometric function: U s µ(x) = (R + 1) s F 1 ( s, d ; d; 4R ), x = R 1. (R + 1) Proof: For x R d+1, x = 1, we write U s µ(x) = x y s dµ(y). S d

6 6 Let x = R, and denote the angle between the two vectors x and y by θ. Then cos θ = x R, y. By the law of cosine, x y = R + 1 R x R, y. Thus by using the Funk-Hecke formula; see Müller [15], we have x y s dµ(y) = (R + 1 R x S d S d R, y ) s/ dµ(y) = ν d 1 ν d 1 1 (R + 1 Rt) s/ (1 t ) (d )/ dt, where ν d denotes the surface area of S d. Using the substitution u = t + 1 and Euler s integral representation of the hypergeometric function F 1, we have 1 (U s µ)(x) = d 1 (R + 1) s ν ( d 1 1 4R ) s/ ν d 0 (R + 1) u u (d )/ (1 u) (d )/ du = d 1 (R + 1) s ν d 1 Γ ( d ) ν d Γ(d) F 1 ( s, d ; d; 4R (R + 1) ). To simplify, we use the formula (see [15]) and then the formula (see [1]) with z = d/. We have ν d = π(d+1)/ Γ( d+1 ), Γ(z) = (π) 1/ z 1/ Γ(z)Γ(z + 1/), (3.4) (U s µ)(x) = d 1 (R + 1) d+1 s Γ( ) Γ d ( ) πγ( d ) Γ(d) F 1 ( s, d ; d; = (R + 1) s F 1 ( s, d ; d; 4R ). (R + 1) 4R (R + 1) ) The following two special cases of Lemma 3 are worth noting. Firstly, when 0 < s < d, the potential (U s µ)(x) is well defined for x = 1. In fact, a simple application of the Lebesgue Dominated Convergence Theorem yields lim U sµ(x) = lim x y s dµ(y) = γ d,s. x 1 x 1 S d On the other hand, when 0 < s < d, the hypergeometric function F 1 ( s, d ; d; z) is continuous at z = 1. Using Equation (.3), we have lim (R + R 1 1) s F 1 ( s, d ; d; 4R (R + 1) ) = s F 1 ( s, d ; d; 1) = s Γ(d)Γ( d s ) Γ(d/)Γ(d s/).

7 7 Thus, we have γ d,s = s Γ(d)Γ( d s ) Γ(d/)Γ(d s/). (3.5) Due to the importance of the constant γ d,s in this paper, we feel reassured that we are able to verify Equation (3.5) directly, and we share the reassurance with the readers. Using Equation (3.4) with z = d/, we write we have s Γ(d)Γ( d s ) Γ(d/)Γ(d s/) Γ(d) = (π) 1/ d 1/ Γ(d/)Γ(d + 1/), = s (π) 1/ d 1/ Γ(d/)Γ((d + 1)/)Γ( d s ) Γ(d/)Γ(d s/) = (π) 1/ d s 1/ Γ( d s d s+1 )Γ( )Γ((d + 1)/) Γ( d s+1 )Γ(d s/) Γ(d s)γ((d + 1)/) = Γ( d s+1 )Γ(d s/). Here in the last step, we have used Equation (3.4) again with z = (d s)/. Secondly, when d =, the potential U s µ(x) has the elementary form: U s µ(x) = 1 R (1 + R) s R 1 s, x = R, s. s Lemma 4. Assume d 1 s < d. Then there exists a positive constant C d,s, independent of N, such that U s µ(x) > γ d,s C d,s N 1+s/d, x = R N,d := 1 + N 1/d. Proof: We first note that with R N,d = 1 + N 1/d, we have (R N,d + 1) s = s (1 s N 1/d ) + (N 1/d ). (3.6) We now estimate F 1 ( s, d ; d; 4R N,d (R N,d ). By using The Fundamental Theorem of Calculus and then Equation (.4), we +1) have F 1 ( s, d ; d; 4R ) N,d (R N,d + 1) = F 1 ( s, d ; d; 1) [ F 1 ( s, d ; d; 1) F 1 ( s, d ; d; = Γ(d)Γ( d s ) Γ(d/)Γ(d s/) s 1 4 4R N,d /(R N,d +1) 4R ] N,d (R N,d + 1) ) F 1 ( s + 1, d + 1; d + 1; z)dz.

8 8 We use Equation (.) to estimate the above integral. For any given ɛ > 0, we have, for N sufficiently large, that F 1 ( s + 1, d + 1; d + 1; z) (1 z) (d s )/ β d,s < ɛ, z [4RN,d /(R N,d + 1), 1], where This implies β d,s = Γ(d + 1)Γ((3d + s)/ + 3). Γ(d/ + 1)Γ(s/ + 1) 1 4R N,d /(R N,d +1) 1 F 1 ( s + 1, d + 1; d + 1; z)dz (β d,s + ɛ) (1 z) (d s )/ dz 4R N,d /(R N,d +1) ( RN,d 1 ) d s = (β d,s + ɛ) d s R N,d + 1 = s d+1 d s (β d,s + ɛ)n 1+s/d. (3.7) Combining Lemma 3 and the above two estimates (3.6) and (3.7), and noting that 1/d 1 s/d, we have U s µ(r N,d ) = s (1 s ( N 1/d Γ(d)Γ( d s ) ) Γ(d/)Γ(d s/) C d,sn 1+s/d) + (N 1+s/d ), which gives the desired result of Lemma 4. In the discussion that follows, we will need the notion of α superharmonic functions and the principal of domination of α superharmonic functions. These topics can be found in Landkof [13]. The definition of α superharmonic functions is technical. Upon checking the pertinent material in Landkof [13], one finds the relation d + 1 α = s between the parameter α used in Landkof [13] and the parameter s we use here. Thus the requirement d 1 s < d translates into 1 α < in Landkof [13]. Furthermore, what is meant in Landkof [13] by an α superharmonic function is in fact a (d + 1 s) superharmonic function here in our context. choose not to use the phrase (d + 1 s) superharmonic function because we feel that notion α superharmonic function has been coined in the mathematical literature. In Chapter 1, Section 5 [13], it is proved that for a σ finite Borel measure λ supported on a compact subset of R d+1, its potential U s λ (0 s < d) is a α superharmonic function. Furthermore, Theorem 1.9 in [13], barring from different notations, states the following result: We

9 9 Theorem 5. Suppose λ is a σ finite positive Borel measure supported on a compact subset of R d+1 whose potential U s λ is finite λ almost everywhere, and that f(x) is an α superharmonic function. If the inequality U s λ(x) f(x) holds λ almost everywhere, then it holds everywhere in R d+1. Theorem 5 is often called the principal of domination for α superharmonic functions. Lemma 6. There exists a constant C d,s, independent of N, such that for all x with x = 1 + N 1/d, U s v N (x) γ d,s C d,s N 1+s/d. (3.8) Proof: By Lemma, we have U s v N (x) γ d,s C d,s N 1+s/d, x = 1. Note that U s µ(x) = γ d,s, x S d, so we can rewrite the above inequality as U s v N (x) U s µ(x)(1 C d,s N 1+s/d ), x = 1. Both measures µ and v N are supported on S d. Since U s v N is an α superharmonic function, we can therefore use Theorem 5 to obtain U s v N (x) U s µ(x)(1 C d,s N 1+s/d ), x R d+1. We then use Lemma 4 to get, for x = 1 + N 1/d, U s v N (x) (γ d,s C d,s N 1+s/d )(1 C d,s N 1+s/d ), which yields the desired estimate. Lemma 7. For every fixed i, (1 i N), we have, for 0 < s < d, N 1 x i x j s γ d,s, x S d. Proof: For every fixed i, (1 i N), since the function x x x j s reaches its minimum at x i on S d, we have N 1 x i x j s N 1 x x j s, x S d.

10 10 Integrating both sides of the above inequality on S d against the probability measure µ(x) yields: N 1 x i x j s N 1 S d x x j s dµ(x) = N 1 N γ d,s, and therefore the desired inequality follows. Theorem 8. The minimum discrete s energy points are well-separated, i.e., there exists a constant C d,s > 0, independent of N, such that min i j x i x j C d,s N 1/d. Proof: Let ω N be a minimal s energy configuration, and let x i0, x j0 be two points in ω N such that x i0 x j0 = min i j x i x j. Using Lemma 7 and then Lemma 6, we have γ d,s N 1 x i0 x j0 s N 1 0 x i0 x j s N 1 x i0 x j0 s = N 1 0,j 0 x i0 x j s N 1 0,j 0 R N,d x i0 x j s N = N 1 R N,d x i0 x j s N 1 R N,d x i0 x i0 s N 1 R N,d x i0 x j0 s j=1 γ d,s C d,s N 1+s/d N 1 R N,d x i0 x i0 s N 1 R N,d x i0 x j0 s. Since for any i, j = 1,,..., N, we have R N,d x i x j N 1/d, it follows that γ d,s N 1 x i0 x j0 s γ d,s (C d,s + )N 1+s/d, which implies that x i0 x j0 C d,s N 1/d. If one follows the trail of the constants throughout the proofs, then one is able to quantitatively estimate the value of the constant C d,s in Theorem 8. However, such a process leads the piling-up of many Gamma function values, among others. There seems to be no obvious way to simplify them. Just for curiosity, we numerically estimate the constant C d,s in Theorem 8 for the case d =, and s = 1. Our numerical result yields C, , putting our estimate of the minimum separation of the

11 11 corresponding minimal energy points on S at / N. The result of Habicht and Van der Waerden [9] for best packing asserts that the maximum diameter of N non-overlapping congruent circles on S is asymptotically ( 8π ) 1/ N N For more information on best packing on S, we also refer readers to [4], [18], and the references therein. References [1] M. Abramowitz and I. Stegun, Handbook of Mathematical Functions, with Formulas, Graphs, and Mathematical Tables, Dover Publications, New York, [] G. E. Andrews, R. Askey, and R. Roy, Special Functions (Encyclopedia of Mathematics and its Applications) Cambridge University Press, Cambridge, [3] J. S. Brauchart, About the second term of the asymptotics for the optimal Riesz energy on the sphere in the potential theoretical case, preprint. [4] J. H. Conway and N. J. A. Sloane, Sphere Packings, Lattices and Groups, nd ed., Springer-Verlag, New York, [5] B. E. J. Dahlberg, On the distribution of Fekete points, Duke Math. J. 45 (1978), [6] S.B. Damelin, P.J. Grabner, Energy functionals, numerical integration and asymptotic equidistribution on the sphere. J. Complexity 19 (003), no. 3, [7] P. D. Dragnev, On the separation of Logarithmic points on the sphere, Approximation Theory X: Abstract and Classical Analysis, Charles K. Chui, Larry L. Schumaker, and Joachim Stöckler (eds.), Vanderbilt University Press, Nashville, TN, 00, pp [8] A. Dubickas, On the maximal product of distances between points on sphere, Lithuanian Math J. 36, no. 3 (1996), [9] W. Habicht and B. L. van der Waerden, Lagerung von Punkten auf der Kugel, Math. Ann. 13 (1951), [10] D. P. Hardin and E. B. Saff, Minimal Riesz energy point configurations for rectifiable d dimsional manifolds, to appear in Adv. Math. [11] D. P. Hardin and E. B. Saff, Discretizing manifolds via minimum energy points, Notices of AMS, Volume 51, Number 10 (004),

12 1 [1] A. B. J. Kuijlaars and E. B. Saff, Asymptotics for minimal discrete energy on the sphere, Trans. Amer. Math. Soc. 350 (1998), [13] N. S. Landkof, Foundation of Modern Potential Theory, Springer-Verlag, Berlin- Heidelberg, 197. [14] A. Martinez-Finkelshtein, V. V. Maymeskul, E. A. Rakhmanov, and E. B. Saff, Asymptotics for minimal discrete Riesz energy on curves in R d, Canad. J. Math. 56 (004), [15] C. Müller, Spherical Harmonics, Lecture Notes in Mathematics, vol 17, Springer Verlag, Berlin-Heidelberg, [16] E.A. Rakhmanov, E.B. Saff, Y.M. Zhou, Minimal discrete energy on the sphere. Math. Res. Lett. 1 (1994), no. 6, [17] E. A. Rakhmanov, E. B. Saff, and Y. M. Zhou, Electrons on the sphere, Computational Methods and Functional Theory, R. M. Ali, St. Ruscheweyh, and E. B. Saff,(eds.), Singpore: World Scientific (1995), pp [18] E. B. Saff and A. B. J. Kuijlaars, Distributing many points on a sphere, Math. Intelligencer 19 (1997), Department of Mathematics Katholieke Universiteit Leuven Celestijnenlaan 00 B B-3001 Leuven (Heverlee) BELGIUM arno@wis.kuleuven.ac.be Department of Mathematics Vanderbilt University Nashville, TN 3740, USA Department of Mathematics Southwest Missouri State University Springfield, MO 65804, USA xis80f@smsu.edu