Determine the global optimizer of the negative-exponent capacity ratio

Determine which compact subsets of Euclidean space maximize the Riesz capacity ratio \(\operatorname{Cap}_q(K)/\operatorname{Cap}_p(K)\) in the parameter regions with negative exponents where the optimizer remains unresolved, including the unresolved regions in one, two, and higher dimensions.

Background

The paper studies the supremum of the ratio of Riesz capacities over compact subsets of Rn\mathbb R^n, focusing on the regime p,q<0p,q<0. Previously established results identify some regions where intervals, two-point sets, balls, or regular simplices are optimal, but substantial parameter regions remain unresolved. The authors combine numerical experiments with partial theoretical results and explicitly distinguish finite computational evidence from proofs of global optimality.

In particular, the unresolved regions include the interval-versus-two-point problem in one dimension, the ball-versus-polygonal configurations in two dimensions, and comparisons involving regular simplices and other finite configurations in higher dimensions.

References

Our aim is to examine the dashed-outlined regions in \autoref{fig:pqdiagram1D}, \autoref{fig:pqdiagram2D} and \autoref{fig:pqdiagram3D}, with $p,q<0$ where the optimizer for the Riesz capacity ratio remains open in one, two, and higher dimensions.

— Riesz capacity ratios with negative exponents  (2609.11186 - Fan, 10 Sep 2026) in Section 8, Summary of results

The two main conjectures we study in one dimension are:

— Riesz capacity ratios with negative exponents  (2609.11186 - Fan, 10 Sep 2026) in Section 1d, subsection “Conjectures,” Conjecture 1d-2pt

We do not have definite conclusions, but if someone in future wants to work on this problem, we hope that the remainder of this section might be useful.

— Riesz capacity ratios with negative exponents  (2609.11186 - Fan, 10 Sep 2026) in Section 2d, subsection “Step 2 — not proved,” Conjecture 3pt

Pólya and Szegő's conjecture extends naturally to all dimensions. We formulate it in terms of maximizing the scale-invariant ratio of capacities.

— Toward Pólya and Szegő's conjecture for logarithmic vs Newtonian capacity  (2609.35438 - Clark et al., 28 Sep 2026) in Section 1, subsection “Higher dimensional Pólya–Szegő conjectures,” Conjecture 1 (labelled \autoref{conj:riesz})