Point Distributions in Compact Metric Spaces

Abstract

Abstract. We consider finite point subsets (distributions) in compact metric spaces. In the case of general rectifiable metric spaces, non-trivial bounds for sums of distances between points of distributions and for discrepancies of distributions in metric balls are given (Theorem 1.1). We generalize Stolarsky's invariance principle to distance-invariant spaces (Theorem 2.1). For arbitrary metric spaces, we prove a probabilistic invariance principle (Theorem 3.1). Furthermore, we construct equalmeasure partitions of general rectifiable compact metric spaces into parts of small average diameter (Theorem 4.1). §1. Introduction. Let M be a compact metric space with a fixed metric θ and a finite non-negative Borel measure µ, normalized by µ(M) = 1. For any metric ρ on M and any N -point subset (distribution) D N ⊂ M, we putand denote by ρ the average value of the metric ρ, given byWe write B r (y) = {x : θ(x, y) < r }, r ∈ T , y ∈ M, for the ball of radius r centred at y and of volume µ(B r (y)). Here T = {r : r = θ (y 1 , y 2 ), y 1 , y 2 ∈ M} is the set of radii, T ⊂ [0, L], where L = sup{r = θ (y 1 , y 2 ) : y 1 , y 2 ∈ M} is the diameter of M in the original metric θ .The local discrepancy of a distribution D N is defined bywhere (B r (y), x) = χ(B r (y), x) − µ(B r (y)), (1.4) and χ (E, x) is the characteristic function of a subset E ⊂ M.

The Stolarsky Principle and Energy Optimization on the Sphere

“…We would also like to mention that the interest in Stolarsky principle in different settings has recently spiked: [13] studied it from the point of view of numerical integration on the sphere, [21] uses it in applications to genomics, [29] explores Stolarsky principle in general metric spaces, [7] connects it to tessellations of the sphere, while the present paper and [5] deal with it in the context of energy optimization.…”

Geodesic distance Riesz energy on the sphere

“…The link between discrepancy and energy on the sphere has been first established by Stolarsky [27] who established an identity relating the spherical cap L 2 discrepancy and the sum of pairwise Euclidean distances between the points of Z. Identities of this type came to be called Stolarsky invariance principle. There has been an increase of activity on this subject in the recent years [7,9,19,22,23,24,5]. In our companion paper [4] we explore a number of variations of this principle and its applications to energy optimization, in particular, part (iii) of Theorem 1.1.…”

Bounds for $$L_p$$-Discrepancies of Point Distributions in Compact Metric Measure Spaces

“…At the same time, the lower bound in (1.7) fails, even for the spheres S d , if the measure ξ is singular. The corresponding example can be found in [5,12,13]. In this example the discrepancy λ p [ξ, N ] is bounded from above by a constant independent of N and p.…”

“…Notice that the upper bound in (1.7) holds for arbitrary compact d-rectifiable spaces M and any measure ξ, see [12]. At the same time, the lower bound in (1.7) fails, even for the spheres S d , if the measure ξ is singular.…”

“…The constants implicit in the symbol are independent of N . The two-side bounds (1.7) were extended to all compact Riemannian symmetric manifolds of rank one (two-point homogeneous spaces) and arbitrary absolutely continuous measures ξ on I, see [13,Theorem 2.2]. Recall that these manifolds are the spheres S d , the real, complex and quaternionic projective spaces FP n , F = R, C, H, and the octonionic projective plane OP 2 , see, for example, [11].…”