7 problems

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 strictly convex $n$-dimensional Minkowski space. Füredi–Lagarias–Morgan conjecture. There exists some constant $\varepsilon>0$ such that … The conjecture would improve…

Let $d4a3^n$ be equipped with the tropical metric, and let $\mathrm{e}_{\mathrm{tr}}(\mathbb{R}^{n})$ denote the maximal cardinality of a tropical equilateral subset of $d4a3^n$. T…

Kusner's conjecture.

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

Let $X$ be a normed vector space of dimension $n$. An equilateral set is a set of points whose pairwise distances are all equal. Equilateral-set conjecture. Every normed $n$-dimens…

Let $X$ be an $n$-dimensional normed space. The equilateral dimension conjecture. … Here $e(X)$ denotes the maximal cardinality of an equilateral set in $X$, that is, a set of pair…