84 problems

Let $X$ be an $n$-dimensional normed space, and let its equilateral dimension be the maximal cardinality of a subset of $X$ whose distinct points all have the same pairwise distanc…

Let $\|\cdot\|$ be a norm on $\mathbb{R}^n$, and define $m_1(\mathbb{R}^n,\|\cdot\|)$ as the supremum of the upper asymptotic densities of measurable unbounded subsets containing n…

Kusner's conjecture.

Erdős's conjecture. There exists $N$ such that no crescent configurations of size $n$ exist for all $n\geq N$.

Global rigidity connectivity conjecture. If $G$ is $2d$-edge-connected and $2$-connected, then $G$ is globally rigid in $X$.

Cauchy–Schwarz conjecture. The Cauchy–Schwarz inequality should hold in every complex normed space. The inequality is established in the paper for some classes, including complex r…

Let $K$ be an $n$-dimensional convex body in $cmathbb{R}^n$. The translative constant volume property means that the volume of the convex hull of any touching pair of translates of…

Swanepoel's conjecture.

Latała's moment conjecture. There exists a universal constant $C$ such that, for every log-concave vector $X$ with values in a finite-dimensional normed space $(F,\|\ \|)$ and ever…

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

Let $X^3$ be a three-dimensional normed space with a differentiable norm, and let $e(X^3)$ denote the largest possible size of an equilateral set in $X^3$. Lawlor–Morgan's conjectu…

Let $V$ be a finite-dimensional Minkowski space, with its norm and Minkowski content defined from the given norm. Gruber's conjecture. The Minkowski content and the norm are propor…

Schäffer's girth duality conjecture. The girth of $X$ equals the girth of $X^*$.

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…

Typical-norm dense-coloring conjecture. For a typical norm,

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…

Let $d4a3^n$ be equipped with the tropical metric, and let $d4a4^n$ denote its integer lattice with the induced tropical metric. The tropical chromatic number conjecture. The tropi…

Let $K\subset\mathbb{R}^2$ be a symmetric convex body with strictly convex $C^1$ boundary. For nonempty compact sets $A,B\subset\mathbb{R}^2$, let $\mathcal{R}_K(a,b)$ be the famil…

Let $\mathbb{M}^3$ be a $3$-dimensional normed space whose unit ball has a $C^2$-class differentiable boundary and strictly positive Gaussian curvature. Existence conjecture. There…

Let $S$ be a bounded set in a $d$-dimensional normed space. Boltyansky–Gohberg conjecture. The set $S$ can be partitioned into at most $2^d$ subsets, each having strictly smaller d…

A $d$-norm is a norm $\|\cdot\|$ on $\mathbb{R}^d$, identified with its unit ball. A set of points is full-dimensional when it is not contained in an affine hyperplane, and a set o…

Let $d\geq 2$, let $n\geq 2d+2$, and let $K_n$ denote the complete graph on $n$ vertices. A graph is edge-redundantly rigid in a normed space if it has a framework that is infinite…

Let $a(G,X)$ denote the algebraic connectivity of a graph $G$ in a normed space $X$, let $K_{2d}$ be the complete graph on $2d$ vertices, and let $T_d$ be the tree defined in the p…

Let $B\in\mathcal{K}^2_o$ be an $o$-symmetric convex body in $\mathbb{R}^2$, let $Q_6(B)$ denote the associated six-vertex convex polygon, and let $B^{\circ}$ be the polar body of…