Logarithmic and Riesz Equilibrium for Multiple Sources on the Sphere --- the Exceptional Case

We consider the minimal discrete and continuous energy problems on the unit sphere $\mathbb{S}^d$ in the Euclidean space $\mathbb{R}^{d+1}$ in the presence of an external field due to finitely many localized charge distributions on $\mathbb{S}^d$, where the energy arises from the Riesz potential $1/r^s$ ($r$ is the Euclidean distance) for the critical Riesz parameter $s = d - 2$ if $d \geq 3$ and the logarithmic potential $\log(1/r)$ if $d = 2$. Individually, a localized charge distribution is either a point charge or assumed to be rotationally symmetric. The extremal measure solving the continuous external field problem for weak fields is shown to be the uniform measure on the sphere but restricted to the exterior of spherical caps surrounding the localized charge distributions. The radii are determined by the relative strengths of the generating charges. Furthermore, we show that the minimal energy points solving the related discrete external field problem are confined to this support. For $d-2\leq s<d$, we show that for point sources on the sphere, the equilibrium measure has support in the complement of the union of specified spherical caps about the sources. Numerical examples are provided to illustrate our results.Comment: 23 pages, 4 figure

Random Point Sets on the Sphere-Hole Radii, Covering, and Separation

Geometric properties of N random points distributed independently and uniformly on the unit sphere (Formula presented.) with respect to surface area measure are obtained and several related conjectures are posed. In particular, we derive asymptotics (as N → ∞) for the expected moments of the radii of spherical caps associated with the facets of the convex hull of N random points on (Formula presented.). We provide conjectures for the asymptotic distribution of the scaled radii of these spherical caps and the expected value of the largest of these radii (the covering radius). Numerical evidence is included to support these conjectures. Furthermore, utilizing the extreme law for pairwise angles of Cai et al., we derive precise asymptotics for the expected separation of random points on (Formula presented.)

Covering of spheres by spherical caps and worst-case error for equal weight cubature in Sobolev spaces

We prove that the covering radius of an N-point subset XN of the unit sphere Sd⊂Rd+1 is bounded above by a power of the worst-case error for equal weight cubature 1N∑x∈XNf(x)≈∫Sdfdσd for functions in the Sobolev space Wps(Sd), where σd denotes normalized area measure on Sd. These bounds are close to optimal when s is close to d/p. Our study of the worst-case error along with results of Brandolini et al. motivate the definition of Quasi-Monte Carlo (QMC) design sequences for Wps(Sd), which have previously been introduced only in the Hilbert space setting p=2. We say that a sequence (XN) of N-point configurations is a QMC-design sequence for Wps(Sd) with s>d/p provided the worst-case equal weight cubature error for XN has order N-s/d as N→∞, a property that holds, in particular, for a sequence of spherical t-designs in which each design has order td points. For the case p=1, we deduce that any QMC-design sequence (XN) for W1s(Sd) with s>d has the optimal covering property; i.e., the covering radius of XN has order N-1/d as N→∞. A significant portion of our effort is devoted to the formulation of the worst-case error in terms of a Bessel kernel, and showing that this kernel satisfies a Bernstein type inequality involving the mesh ratio of XN. As a consequence we prove that any QMC-design sequence for Wps(Sd) is also a QMC-design sequence for Wp's(Sd) for all 1≤ps'>d/p