49 problems

Böröczky's conjecture. There is a point $x\in P$ with norm

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…

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…

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…

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…

Pinned most-norms conjecture. For most $d$-norms $\|\cdot\|$, every finite point set $P\subseteq\mathbb R^d$ contains a point $x\in P$ determining $(1-o(1))|P|$ distances to the ot…

Hilbert-space characterization conjecture. The following conditions are equivalent: $\mathbb{X}$ is a Hilbert space; and a nonempty set $A\subset S_{\mathbb{X}}$ is a minimal…

Let $K^{\lor}$ be the spherical completion of the underlying valued field $K$, let $V_K$ denote the relevant multiplicative value subgroup, and let $\sim$ denote the equivalence re…

Let $F4D0\mathcal{F}$ be an NS-family of $n\geq 1$ translates of an origin-symmetric convex domain $F4D0\mathbf{M}$. Then … This bound generalizes the preceding successive-perimete…

Let $V$ be a finite-dimensional real vector space, let $2\leq k<\dim V$, and let $K\subset V$ be a convex body. Suppose that all $k$-dimensional sections of $K$ through a fixed int…

Let $V$ be a normed space over the reals, and let $2\leq k<\dim V$ be an integer. Suppose that all linear $k$-dimensional subspaces of $V$ are isometric to each other. Banach's iso…

Block-and-hole minimal-rigidity conjecture. The following statements are equivalent:

Let $E=(\mathbb{R}^n,\left\Vert\cdot\right\Vert)$ be a hypermetric normed space, and let $K\in\mathcal{K}^n$, the family of compact convex bodies in $\mathbb{R}^n$. Let…

Existence conjecture. There exists a normed space with a second-order rigid framework that is locally and continuously flexible.

Let $B(n)$ be the maximum, over all norms on $\mathbb{R}^n$, of the Borsuk number of $(\mathbb{R}^n,\left\Vert\cdot\right\Vert)$, where the Borsuk number is the smallest number of…

Kusner's conjecture.