9 problems

For $s\in(2,4)$, let ${\mathpzc E}_s(\mathbb{S}^2,N)$ be the minimal total Riesz $s$-energy, let $W_s$ be the analytically continued continuum coefficient, and let…

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…

For $-2<s<2$ with $s\neq0$, let $u_s(N)$ be the minimal average unstandardized Riesz pair-energy on $\mathbb{S}^2$, let $W_s$ denote the continuum energy, and let $C_s$ be a consta…

Let $v_s(N)$ be the minimal average standardized Riesz pair-energy on $\mathbb{S}^2$, and let $\ddot v_s(N)=v_s(N-1)-2v_s(N)+v_s(N+1)$. Let $s^\dagger=15.04807\ldots$, and assume t…

Let $v_s(N)$ be the minimal average standardized Riesz pair-energy on $\mathbb{S}^2$, and let ${\cal C}_+(s)$ be the set of integers $N$ at which strict local concavity fails, so t…

Minus-one concavity conjecture. The map $N\mapsto v_{-1}(N)$ is locally strictly concave for all $N>2$, equivalently $\ddot v_{-1}(N)<0$ for all $N>2$. Numerical data suggest this…

Let $d\geq1$, let $\mathbb{S}^d$ be the unit sphere, and let $\mathcal{H}_d(\mathbb{B}^d)$ and $\mathcal{H}_d(\mathbb{S}^d)$ denote the relevant measures of the unit ball and spher…

Let $d\geq 2$, let $\mathbb{S}^d$ be the unit sphere, let $\mathcal{H}_d(\mathbb{S}^d)$ denote its $d$-dimensional Hausdorff measure, and let $V_s(\mathbb{S}^d)$ denote its continu…

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…