11 problems

Let $c_n(X)$ denote the $n^{\text{th}}$ linear polarization constant of a Banach space $X$, defined by … Here ${\mathbb R}^n$ is the real Hilbert space of $n$-tuples with the Eucli…

Let $c_n(X)$ denote the $n$-th linear polarization constant of a Banach space $X$, and let ${\mathbb R}^n$ be the $n$-dimensional real Euclidean space. Conjecture. … This conjectur…

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

Strong polarization extremizer classification conjecture. All extremal cases of the strong polarization problem can be obtained by iterated orthogonal sums starting from point sets…

Planar extremizer iteration conjecture. If the initial set is not extremal, then successive iteration of the map $\Phi$ converges toward the essentially unique extremal case, throu…

Extremizer uniqueness conjecture. The only extremal vector system in the real polarization problem is the orthonormal system consisting of $n$ unit vectors in $\mathbb{R}^n$.

Strong polarization problem. There exists a unit vector $v\in\mathbb{R}^d$ such that

Arias-de-Reyna's conjecture. The best possible constant for real Hilbert spaces agrees with the best possible constant for complex Hilbert spaces; equivalently, for every real Hilb…

Limit-existence conjecture. The limit always exists, so 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…