11 problems

A finite point set $S$ in Euclidean space admits standard Delaunay triangulations, and the mean radius functional and the $D$ functional are functionals defined on such triangulati…

KPZ fluctuation exponent conjecture. The right order of the fluctuations of $T(\0,{\mathbf n})-\mu n$ is $n^{\chi}$, where

Let the dilation of a geometric graph be the maximum, over pairs of vertices, of the ratio between the shortest-path length in the graph and their Euclidean distance. Chew's conjec…

Order-$k$ Delaunay optimality conjecture. The hypertriangulation $P_k$ has the maximum number of triangles if and only if $P_k$ is the order-$k$ Delaunay triangulation of $A$.

Uniqueness conjecture. The order-$k$ Delaunay triangulation is the only level-$k$ hypertriangulation with maximally many triangles that has the local angle property.

Let $\mathcal{C}\subset\mathbb{R}^2$ be an infinite point configuration with unit density, and let $f$ be completely monotone with $|f(r)|=O(r^{-1-\varepsilon})$ as $r\to\infty$ fo…

Given a point set in the plane, its $L_2$-Delaunay graph is the classical Euclidean Delaunay graph. The stretch of a plane spanner is the maximum, over pairs of vertices, of the ra…

Nearly quadratic kinetic Delaunay conjecture. The maximum possible number of discrete changes experienced by the Euclidean Delaunay triangulation during such a motion is nearly qua…

Let $L_0$ denote the relevant geometric walk-length parameter and let $omega_n$ denote the paper's parameter controlling the random Delaunay triangulation. The algorithm…

The algorithm $proc{Simple-Path}$ generates a path in the Delaunay triangulation $\operatorname{DT}(\mathbf{X})$; call it competitive when its expected length satisfies the same as…

General-dimensional extension conjecture. These results should extend to the general setting of arbitrary $n$ and $N\geq n$.