18 problems

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

Strip energy ball comparison conjecture. If

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…

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 $\omega_N^{\log}$…

Hypersingular triangular-lattice expansion conjecture. For $2<s<4$,

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 $N\geq 2$, 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 $s\in(2,4)$, 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 $d\geq1$, 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

The discrepancy asymptotic conjecture. If the fundamental conjecture for optimal Riesz energy on spheres holds, then

Let $C_{s,d}$ be the asymptotic Riesz-energy constant defined, for $d\geq 2$, by … Here $\mathcal{E}_s([0,1]^d;N)$ denotes the optimal discrete Riesz $s$-energy of $N$ points in th…

Let $d\geq 2$, let $-2<s<d+2$ with $s\neq 0,d$, and let $\mathbb{S}^d$ be the unit sphere in $\mathbb{R}^{d+1}$. Write $\mathcal{E}_s(\mathbb{S}^d;N)$ for the optimal discrete Ries…