28 problems

Brauchart–Hardin–Saff conjecture. The following asymptotic expansion holds:

Let $C_{s,d}$ be the asymptotic Riesz $s$-energy constant for $d$-rectifiable sets, and let … be the Epstein zeta function of a lattice $\Lambda\subset\mathbb{R}^d$ with covolume…

Strip energy ball comparison conjecture. If

Let $d\geq1$ and $\max(0,d-2)<s<d$. Let $\overline{\mathbb B}$ be the closed unit ball, let $\mathbb S$ be its boundary sphere, and let $S_{Q,\overline{\mathbb B}}$ denote the supp…

Consider the weakly admissible equilibrium problem in $\mathbb{R}^d$ with $d-2\leq s<d$ and an attractor–repellent external field generated by points $y_1=(0;y_{1,d+1})$ and…

Maximal separation conjecture. The points are maximally separated in the sense that

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 $\mathcal{E}_s(N)$ be the minimal Riesz $s$-energy of an $N$-point configuration on $\mathbb{S}^2$, and let $\sigma$ be normalized surface area measure. For $-2<s<2$, $s\neq 0$…

Let $cmathbb{S}^d$ be the unit sphere and let $n=cpi_L$. Let $K$ be an isotropic kernel producing $n$ points through its determinantal point process, and let $E_s(x)$ denote the Ri…

Let $cmathcal{E}(s,n)$ denote the minimal Riesz $s$-energy of $n$ points on $cmathbb{S}^d$, let $V_s(cmathbb{S}^d)$ be the continuous Riesz $s$-energy of the sphere, and let…

Energy lower-limit conjecture. The following inequalities hold:

Let $\mathcal{V}_2$, $\mathcal{V}_4$, $\mathcal{V}_8$, and $\mathcal{V}_{24}$ denote respectively the equilateral triangular, $D_4$, $E_8$, and Leech lattices, and let…

Let $N\geq 2$, let $\alpha\in(0,N)$, and let $I_{\alpha}(K)$ denote the $\alpha$-Riesz energy of a convex body $K\subset\mathbb{R}^N$. For $r>0$, consider convex bodies with perime…

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 $d\geq1$, 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…

The spherical discrepancy constant conjecture. If both the fundamental optimal Riesz-energy conjecture and the Kuijlaars--Saff conjecture hold, then