[Paper Review] Optimal discrete measures for Riesz potentials

Sergiy Borodachov, Douglas P. Hardin+2 more|arXiv (Cornell University)|Jun 13, 2016

Spectral Theory in Mathematical Physics18 references3 citations

TL;DR

This paper establishes the asymptotic behavior of optimal N-point discrete measures for Riesz potentials with $ s \geq d $ on d-rectifiable subsets of $ \mathbb{R}^p $, deriving the dominant term of the N-point Riesz s-polarization constant as $ N \to \infty $. It proves that the weak-star limit distribution of asymptotically optimal configurations is proportional to the weighted Hausdorff measure $ \mathcal{H}_d^{s,w} $, with explicit asymptotic constants involving the surface area and weight function.

ABSTRACT

For $s\geqslant d$, we obtain the leading term as $N o \infty$ of the maximal weighted $N$-point Riesz $s$-polarization (or Chebyshev constant) for a certain class of $d$-rectifiable compact subsets of $\mathbb{R}^p$. This class includes compact subsets of $d$-dimensional $C^1$ manifolds whose boundary relative to the manifold has $\mathcal{H}_d$-measure zero, as well as finite unions of such sets when their pairwise intersections have $\mathcal{H}_d$-measure zero. We also explicitly find the weak$^*$ limit distribution of asymptotically optimal $N$-point polarization configurations as $N o \infty$.

Motivation & Objective

• To determine the asymptotic behavior of the N-point Riesz s-polarization constant for compact d-rectifiable sets in $ \mathbb{R}^p $ when $ s \geq d $.
• To characterize the weak-star limit distribution of asymptotically optimal N-point configurations as $ N \to \infty $.
• To establish the dominant term in the asymptotic expansion of the maximal polarization constant for weighted Riesz potentials on manifolds and finite unions of such sets.
• To prove that the optimal configuration distribution converges to a measure proportional to $ \mathcal{H}_d^{s,w} $, the weighted d-dimensional Hausdorff measure.

Proposed method

• Uses a discretization scheme based on dyadic cubes and Whitney-type covers to approximate the set $ A $, ensuring small boundary measure and controlled overlap.
• Applies the Lebesgue Dominated Convergence Theorem to the sequence of characteristic functions of cubes, enabling convergence of weighted sums to integrals over the set.
• Employs a comparison argument via the function $ u_n(x) = \sum_{\Gamma \in \mathfrak{G}_n} \overline{w}_\Gamma^{-d/s} \chi_\Gamma(x) $ to relate discrete configurations to continuous measures.
• Derives lower and upper bounds for the polarization constant using estimates on the number of points in subsets $ \Gamma \subset A $, leveraging the parameter $ \tau_{s,d}(N) $, the maximal number of points in a set of diameter $ \sim N^{-1/d} $.
• Uses the fact that $ \lim_{N \to \infty} \#(\omega_N \cap B)/N = \mathcal{H}_d^{s,w}(B)/\mathcal{H}_d^{s,w}(A) $ for Borel sets $ B \subset A $, proving weak-star convergence of the empirical measures to $ \mathcal{H}_d^{s,w} $.
• Combines lower and upper bounds for $ \overline{h}^{w}_{s,d}(A) $ to show equality, completing the proof of the asymptotic constant.

Experimental results

Research questions

• RQ1 What is the asymptotic growth rate of the N-point Riesz s-polarization constant for $ s \geq d $ on d-rectifiable sets?
• RQ2 How do the optimal N-point configurations distribute as $ N \to \infty $ on such sets?
• RQ3 What is the precise limiting measure that governs the distribution of optimal configurations?
• RQ4 Can the polarization constant be expressed in terms of the weighted Hausdorff measure $ \mathcal{H}_d^{s,w} $?
• RQ5 Under what conditions does the weak-star limit of empirical measures of optimal configurations exist and equal $ \mathcal{H}_d^{s,w} $?

Key findings

• The dominant term of the N-point Riesz s-polarization constant as $ N \to \infty $ is $ \left( \frac{\sigma_{s,d}}{\mathcal{H}_d^{s,w}(A)^{s/d}} \right) N^{s/d} $, where $ \sigma_{s,d} $ is a geometric constant.
• The weak-star limit distribution of asymptotically optimal N-point configurations is proportional to the weighted Hausdorff measure $ \mathcal{H}_d^{s,w} $, with normalization $ \mathcal{H}_d^{s,w}(A)^{-1} \mathcal{H}_d^{s,w} $.
• For any Borel subset $ B \subset A $, the proportion of points in $ \omega_N \cap B $ converges to $ \mathcal{H}_d^{s,w}(B)/\mathcal{H}_d^{s,w}(A) $ as $ N \to \infty $.
• The asymptotic constant $ \overline{h}^{w}_{s,d}(A) $ satisfies $ \overline{h}^{w}_{s,d}(A) = \sigma_{s,d} / \mathcal{H}_d^{s,w}(A)^{s/d} $, matching the lower bound from earlier estimates.
• The result holds for compact d-rectifiable sets with $ \mathcal{H}_d $-null boundary relative to the manifold and for finite unions of such sets with measure-zero intersections.
• The proof relies on a delicate balance between cube partitioning, measure control, and convergence of weighted sums, culminating in a sharp asymptotic formula.