Riesz s-Equilibrium Measures on d-Rectifiable Sets as s Approaches d

Abstract

Abstract. Let A be a compact set in R p of Hausdorff dimension d. For s ∈ (0, d), the Riesz s-equilibrium measure µ s is the unique Borel probability measure with support in A that minimizesover all such probability measures. If A is strongly (H d , d)

Optimal $$N$$ N -Point Configurations on the Sphere: “Magic” Numbers and Smale’s 7th Problem

“…Yet it can be understood by considering a certain limit process s → 2; cf. [CaHa09] for the limit process s → d in analogous optimization problems formulated on d-dimensional manifolds. The Riesz pair interaction for s = 2, in physics considered as correction term to Newton's gravity [Man25], is also special in 36 An important special case of Quasi-Monte Carlo schemes are the so-called t-designs, which can be characterized by polynomial energy functionals [SlWo09].…”

Riesz s-equilibrium measures on d-dimensional fractal sets as s approaches d

“…One can see µ s, [−1,1] converges in the weak-star topology on M(A) to H d A /H d (A) as s ↑ 1. More generally it is shown in [2] that this convergence occurs for certain d-rectifiable sets.…”

Distributing many points on spheres: Minimal energy and designs

“…We remark that the case s = d for S d has already been dealt with in [100] using results from [136]. Furthermore, in [66] it was shown that the s-equilibrium measures on Ω converge to the normalised d-dimensional Hausdorff measure restricted to Ω as s → d − under rather general assumptions on Ω. We remark that I. Pritzker [165] studied the discrete approximation of the equilibrium measure on a compact set Ω ⊂ R p , p ≥ 2, with positive s-capacity by means of points which do not need to lie inside Ω.…”