15 problems

Let $S^2$ be the unit sphere. For integers $m,n$, consider the distribution of … points on $S^2$ obtained from the natural embedding of the icosahedron, with the points arranged in…

Support conjecture for the causal variational principle. All minimizers of this energy are discrete whenever $\tau>\sqrt{2}$; furthermore, there exist discrete minimizers for…

For each $s$, let ${\cal C}_+(s)$ be the set of $N$-values at which the actual map $N\mapsto v_s(N)$ is strictly convex, equivalently where local strict concavity fails. Defect-set…

Let $v_s^x(N)$ be the empirically computed minimal average standardized Riesz pair-energy for $N$ points on $\mathbb{S}^2$. Empirical concavity-threshold conjecture. There exists…

Let ${E}_2(N)$ be the minimal total Riesz $2$-energy of $N$ points on the sphere, and let $\gamma$ and $\gamma_1$ denote the Euler constant and the first generalized Stieltjes cons…

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…

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 $v_s(N)$ denote the minimal average standardized Riesz pair-energy of $N$ points on the circle $\mathbb{S}^1$, and let local strict concavity mean $v_s(N-1)-2v_s(N)+v_s(N+1)<0$…

Let $\mathcal E_3(\mathbb S^2,N)$ denote the minimal Riesz $3$-energy of $N$ points on the unit sphere. The empirical second-order conjecture. The second-order term is conjectured…

Let $\mathcal E_s(\mathbb S^2,N)$ be the minimal Riesz $s$-energy of $N$ points on the unit sphere, with $s>2$. Let $\Lambda$ be the unit hexagonal lattice, and let…

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…