We derive fundamental asymptotic results for the expected covering radius $rho(X_N)$ for $N$ points that are randomly and independently distributed with respect to surface measure on a sphere as well as on a class of smooth manifolds. For the unit sp
here $mathbb{S}^d subset mathbb{R}^{d+1}$, we obtain the precise asymptotic that $mathbb{E}rho(X_N)[N/log N]^{1/d}$ has limit $[(d+1)upsilon_{d+1}/upsilon_d]^{1/d}$ as $N to infty $, where $upsilon_d$ is the volume of the $d$-dimensional unit ball. This proves a recent conjecture of Brauchart et al. as well as extends a result previously known only for the circle. Likewise we obtain precise asymptotics for the expected covering radius of $N$ points randomly distributed on a $d$-dimensional ball, a $d$-dimensional cube, as well as on a 3-dimensional polyhedron (where the points are independently distributed with respect to volume measure). More generally, we deduce upper and lower bounds for the expected covering radius of $N$ points that are randomly and independently distributed on a metric measure space, provided the measure satisfies certain regularity assumptions.
We consider periodic energy problems in Euclidean space with a special emphasis on long-range potentials that cannot be defined through the usual infinite sum. One of our main results builds on more recent developments of Ewald summation to define th
e periodic energy corresponding to a large class of long-range potentials. Two particularly interesting examples are the logarithmic potential and the Riesz potential when the Riesz parameter is smaller than the dimension of the space. For these examples, we use analytic continuation methods to provide concise formulas for the periodic kernel in terms of the Epstein Hurwitz Zeta function. We apply our energy definition to deduce several properties of the minimal energy including the asymptotic order of growth and the distribution of points in energy minimizing configurations as the number of points becomes large. We conclude with some detailed calculations in the case of one dimension, which shows the utility of this approach.
Let $G$ be a finite union of disjoint and bounded Jordan domains in the complex plane, let $mathcal{K}$ be a compact subset of $G$ and consider the set $G^star$ obtained from $G$ by removing $mathcal{K}$; i.e., $G^star:=Gsetminus mathcal{K}$. We refe
r to $G$ as an archipelago and $G^star$ as an archipelago with lakes. Denote by ${p_n(G,z)}_{n=0}^infty$ and ${p_n(G^star,z)}_{n=0}^infty$, the sequences of the Bergman polynomials associated with $G$ and $G^star$, respectively; that is, the orthonormal polynomials with respect to the area measure on $G$ and $G^star$. The purpose of the paper is to show that $p_n(G,z)$ and $p_n(G^star,z)$ have comparable asymptotic properties, thereby demonstrating that the asymptotic properties of the Bergman polynomials for $G^star$ are determined by the boundary of $G$. As a consequence we can analyze certain asymptotic properties of $p_n(G^star,z)$ by using the corresponding results for $p_n(G,z)$, which were obtained in a recent work by B. Gustafsson, M. Putinar, and two of the present authors. The results lead to a reconstruction algorithm for recovering the shape of an archipelago with lakes from a partial set of its complex moments.
We study reverse triangle inequalities for Riesz potentials and their connection with polarization. This work generalizes inequalities for sup norms of products of polynomials, and reverse triangle inequalities for logarithmic potentials. The main to
ol used in the proofs is the representation for a power of the farthest distance function as a Riesz potential of a unit Borel measure.
We study the reverse triangle inequalities for suprema of logarithmic potentials on compact sets of the plane. This research is motivated by the inequalities for products of supremum norms of polynomials. We find sharp additive constants in the inequ
alities for potentials, and give applications of our results to the generalized polynomials. We also obtain sharp inequalities for products of norms of the weighted polynomials $w^nP_n, deg(P_n)le n,$ and for sums of suprema of potentials with external fields. An important part of our work in the weighted case is a Riesz decomposition for the weighted farthest-point distance function.
For a closed subset $K$ of a compact metric space $A$ possessing an $alpha$-regular measure $mu$ with $mu(K)>0$, we prove that whenever $s>alpha$, any sequence of weighted minimal Riesz $s$-energy configurations $omega_N={x_{i,N}^{(s)}}_{i=1}^N$ on $
K$ (for `nice weights) is quasi-uniform in the sense that the ratios of its mesh norm to separation distance remain bounded as $N$ grows large. Furthermore, if $K$ is an $alpha$-rectifiable compact subset of Euclidean space ($alpha$ an integer) with positive and finite $alpha$-dimensional Hausdorff measure, it is possible to generate such a quasi-uniform sequence of configurations that also has (as $Nto infty$) a prescribed positive continuous limit distribution with respect to $alpha$-dimensional Hausdorff measure. As a consequence of our energy related results for the unweighted case, we deduce that if $A$ is a compact $C^1$ manifold without boundary, then there exists a sequence of $N$-point best-packing configurations on $A$ whose mesh-separation ratios have limit superior (as $Nto infty$) at most 2.
We investigate the minimal Riesz s-energy problem for positive measures on the d-dimensional unit sphere S^d in the presence of an external field induced by a point charge, and more generally by a line charge. The model interaction is that of Riesz p
otentials |x-y|^(-s) with d-2 <= s < d. For a given axis-supported external field, the support and the density of the corresponding extremal measure on S^d is determined. The special case s = d-2 yields interesting phenomena, which we investigate in detail. A weak* asymptotic analysis is provided as s goes to (d-2)^+.
