10 problems

Real polarization conjecture. The displayed existence and equality characterization hold for the stated unit vectors. The surrounding text says that the corresponding complex quest…

Strong polarization conjecture. The inequality above holds for every set of $n$ vectors in $\mathbb{S}^{d-1}$.

Let $\mathbb{S}^1$ be the unit circle, let $\omega_N^*$ denote a configuration of $N$ equally spaced points on it, and let $P_s(\omega_N^*)$ be its Riesz polarization. For…

Let $f_t^{(n)}(\vec{x})$ be the empirical distribution of $n$ agents in the unit square under the two-dimensional Attraction-Repulsion Model, with interaction threshold $\tau$. A p…

Let $f_t^{(n)}$ be the empirical distribution of the population in the one-dimensional Attraction-Repulsion Model, and let $\tau$ be the interaction threshold. For parameters…

The hexagonal lattice conjecture. The optimizer of this polarization problem is the hexagonal lattice $A_2$. The existence of optimizers is noted under rapid decay assumptions ensu…

Let $s>0$, let $\mathbb S^{p-1}\subset\mathbb R^p$ be the unit sphere, and consider the unconstrained polarization problem $\mathcal P_s^*(\mathbb S^{p-1},N)$, in which the $N$ poi…

The hexagonal-lattice conjecture. The planar polarization constant should satisfy

Let $E$ be a compact set of positive capacity, let $\mu_E$ be its equilibrium measure, and let $d_E(x):=\sup_{t\in E}|x-t|$ be its farthest distance function. Define … A Pritsker–R…

Let ${\Bbb S}^1$ be the unit circle, let $p>0$, and let … be the set of $n$-th roots of unity. For a configuration $\omega_n\subset {\Bbb S}^1$, write $M^p(\omega_n,{\Bbb S}^1)$ fo…