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 $\mathbb R^n$, focusing on the regime $p,q<0$. 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.