4 problems

Nonnegativity conjecture for Riesz kernels. For $\alpha\gg 0$, the Riesz kernel $q_\alpha(\sigma)$ takes nonnegative values on $K$.

Let $\mathbb{S}^1$ be the unit circle, let $\omega_N^*$ denote a configuration of $N$ equally spaced points on it, and let $P_s(\omega_N^*)$ be its Riesz polarization. For…

Radial arcsine conjecture. The radial arcsine distribution solves the full-space equilibrium problem in $\mathbb{R}^d$ for all dimensions $d$.

The hexagonal-lattice conjecture. The planar polarization constant should satisfy