6 problems

Let $\omega_N\subset\mathbb{S}^2$ be a collection of $N$ antipodal points, meaning that $p\in\omega_N$ implies $-p\in\omega_N$. Let $\mathscr{E}_{\log}(\omega_N)$ denote its logari…

Optimal-measure claim. For any fixed $n\geq 1$, the optimal measure is $d\sigma=w^\lambda dx$ for some $\lambda\in(-1/2,\infty)$. The value of $\lambda$ may depend on $n$.

Let $E_s(\omega_N)$ denote the Riesz $s$-energy of a configuration $\omega_N\subset\mathbb{S}^2$, let $E_{\log}(\omega_N)$ denote its logarithmic energy, and let $\omega_N^{\log}$…

Let $D(\omega_N)$ denote the spherical cap discrepancy of a sequence of configurations $\{\omega_N\}_{N=2}^{\infty}\subset\mathbb{S}^2$. Call the sequence well-separated when its p…

Let $u_0(N)$ be the minimal average logarithmic pair-energy of $N$ points on $\mathbb{S}^2$, let $W_{\mathrm{log}}$ be its continuum contribution, and let $C_{\mathrm{log}}$ be the…

Let $\mathcal{E}_{\mathrm{log}}(\mathbb{S}^2;N)$ denote the optimal logarithmic energy of $N$ points on the unit sphere, and let $C_{\mathrm{log},2}$ and $D_{\mathrm{log},2}$ be co…