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 provide an introduction to logarithmic potential theory in the complex plane that particularly emphasizes its usefulness in the theory of polynomial and rational approximation. The reader is invited to explore the notions of Fekete points, logarit
hmic capacity, and Chebyshev constant through a variety of examples and exercises. Many of the fundamental theorems of potential theory, such as Frostmans theorem, the Riesz Decomposition Theorem, the Principle of Domination, etc., are given along with essential ideas for their proofs. Equilibrium measures and potentials and their connections with Green functions and conformal mappings are presented. Moreover, we discuss extensions of the classical potential theoretic results to the case when an external field is present.
