13 problems

Let $\alpha(d,k,\mathbb{F})$ denote the maximum size of a $k$-nearly orthogonal set in $\mathbb{F}^d$. Over the real field, Füredi and Stanley showed that…

Weyl–Heisenberg covariant SIC-POVM conjecture. There exists a normalized $\lvert\phi\rangle\in\mathbb{C}^d$ such that the set $\{D_{jk}\lvert\phi\rangle\}_{j,k=1}^{d}$ is a SIC-POV…

A SIC-POVM is a set $P$ of $d^{2}$ normalized vectors $\lvert\phi_k\rangle$ in $\mathbb{C}^d$ satisfying … Its POVM elements are the subnormalized projectors…

Let $N$ be a positive integer and let $E$ be a subspace of $l^N_{\infty}$. For a positive integer $k$, write $l^k_{\infty}$ for the $k$-dimensional coordinate space, and let…

Equilateral number conjecture. Every $n$-dimensional normed space $X$ satisfies

Let $S$ be a $k$-distance set in an $n$-dimensional normed space, meaning that the set of distances between distinct points of $S$ contains at most $k$ numbers. $k$-distance set co…

Let $X$ be an $n$-dimensional normed space, and let $e(X)$ denote the maximum cardinality of an equilateral subset of $X$. Lower-bound conjecture. If $\dim X=n$, then … This is the…

Let $X$ be a finite-dimensional Banach space, with $n=\dim X$. Finite-dimensional characterization conjecture. (a) If every one-dimensional subspace of $X$ can be represented as an…

Gerzon's equality conjecture. The only dimensions in which equality in Gerzon's bound is attained are

$4\times8$ uniqueness conjecture. There exists a unique complex $4\times8$ ETF up to switching equivalence.

For a finite-dimensional Banach space $F$, let $d_2(F)$ denote its Banach–Mazur distance to the Euclidean space of the same dimension. A Banach space is $d_2$-minimal if it has the…

Let $\beta_4$ denote the parameter in the four-dimensional version of Beck's theorem, where a set of $n$ points is required either to determine many hyperplanes or to have a hyperp…

Let $N$ lines in $\mathbb{R}^3$ form an optimal line packing, meaning that their minimum angle is as large as possible among configurations of $N$ lines. Choosing one unit vector f…