11 problems

Let $({\mathbb S}^2)^n$ be the configuration space of $n$ points on the unit sphere, and let $\mathcal{E}_{\log}(n)$ denote the minimum of … over $({\mathbb S}^2)^n$, where…

Let $\mathbb S^2=\{x\in\mathbb R^3:\|x\|=1\}$, and for points $x_1,\ldots,x_n\in\mathbb S^2$ define the discrete logarithmic energy … Let $C_{\sf{min}}$ be the constant in the asym…

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…

Sharpness conjecture. The upper bound is an equality:

Sharpness conjecture. The upper bound is an equality:

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 $2\le k\le n$, and consider configurations of $n+k$ points in ${\Bbb S}^{n-1}$. A collection of simplices is mutually orthogonal when the linear spans of distinct simplices are…

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 $d$ be a positive integer, and let $[x]$ denote the greatest integer function. Consider configurations of $d+2$ points on the unit sphere $\mathbb{S}^{d-1}$, and let…

Let $W$ be the renormalized energy on discrete subsets of $\mathbb R^2$, and interpret asymptotic density one in the usual planar sense. Sandier–Serfaty conjecture. The triangular…