3 problems

Ellipsoid–hyperboloid conjecture. If the diagonal of $M$ is $\lambda\mathbf{1}$ and the ellipsoid $\mathcal{E}$ meets every branch of $\mathcal{H}$, then $\lambda\leqslant1$.

Inverse-eigenvector product conjecture. There exists an inverse eigenvector $\alpha$ such that

Let $d\geq 2$, let $x_1,\ldots,x_d$ be unit vectors in $\mathbb{R}^d$, and let $v$ be a unit vector in $\mathbb{R}^d$. The $d$th linear polarization constant $c_d(\mathcal{H})$ is…