8 problems

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…

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…

Graphical distinct-realisation conjecture. For every $\varepsilon>0$, there exists a positive constant $C_\varepsilon$, independent of $P$, such that

Weak pinned distance conjecture. For every $\varepsilon>0$, there exists a constant $C_\varepsilon>0$ such that every finite point set $P\subseteq\mathbb R^2$ contains a point…

Distinct-distances conjecture. For every such pair of sets, $P_1\times P_2$ spans $\Omega(n^{3/2})$ distinct distances, unless each of $\gamma_1$ and $\gamma_2$ is an algebraic hel…

Let $Y_g$ be a standard regular hyperbolic surface of genus $g\geq 2$, and let $K_{Y_g}$ denote its geodesic-covering number. Geodesic-covering conjecture for standard regular surf…

Let $\mathcal P$ be a set of $n$ points in $\mathbb R^2$, and let $D(\mathcal P)$ denote the number of distinct distances determined by $\mathcal P$. No-heavy-curves conjecture. If…

Let $P$ be a set of $N$ points in the plane, and let $\delta(N)$ denote the largest size of a subset of $P$ whose pairwise distances are all distinct. For every $\epsilon>0$, conside…