6 problems

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…

$4\times8$ 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…

Petty's conjecture. For all normed spaces $X$ of dimension $n$,