Ball–Frenkel strong polarization conjecture

Let $v_1,\dots,v_n\in\mathbb{S}^{d-1}$ be a set of unit vectors. Then there exists a unit vector $u\in\mathbb{S}^{d-1}$ such that

$$\sum_{j=1}^n\frac{1}{\langle v_j,u\rangle^2}\leq n^2.$$

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

The abstract states that the authors prove this conjecture, so it is solved; the paper also reports new extremizers arising from Coxeter systems.