100 problems

Let $Z_{{o}}$ be the zero cell, meaning the cell containing the origin, in a planar stationary isotropic Poisson line tessellation, and let $V_2(Z_{{o}})$ denote its area. Kendall'…

Equally spaced configuration conjecture. The configuration attaining maximal lifetime consists of $m$ points equally spaced on the unit circle.

Critical logarithmic CLT conjecture. Let $d\in\mathbb{N}$. There exists a constant $σ_{d,d/2}^2\in(0,\infty)$ such that

Penrose–Ong conjecture. Suppose $d\in\mathbb{N}$. The limit theorems for $\d835α\in(0,d/4)$, namely the variance limits and the corresponding Gaussian convergence for the binomial…

Uniqueness and dynamical characterization conjecture. The isometry-invariant thermodynamic limit ${\cal A}_{\mathbb{R}^3}$ should be unique, and should coincide with the polyhedral…

Conjecture 1. For every ${\rm c}\in(-\infty,1]$, there exists a unique, up to multiplication by a positive constant, non-zero ${\rm c}$-LCC assignment…

Zaporozhets–Tarasov conjecture. For every convex body $K$ in $\mathbb{R}^d$,

Let $d\geq 1$, let $\beta>\frac d2$, and let $\widetilde K_n^\beta$ be the random beta-prime polytope in $\mathbb{R}^d$. For $s\in\{1,\ldots,d\}$, write $V_s(\widetilde K_n^\beta)$…

Let $Z_{b3,\lambda,d}^{(\lambda)}$ denote the visibility region in hyperbolic $d$-space obtained by observing along a $\lambda$-geodesic ray through a Poisson process of obstructin…

Conditional non-degeneracy conjecture. For fixed $k\geq 2$, the conditional distribution of $\alpha(\mathcal W_{d+k,d})$ converges weakly as $d\to\infty$ to a limit law not concent…

Higher-moment asymptotics. For every fixed $k\in \{2,3,\ldots\}$, we have

Let $(\mathcal{T}_n)_{n\in\mathbb{N}}$ be stationary random tessellations in $\mathbb{R}^d$ converging to a stationary random tessellation $\mathcal{T}$, and let…

Consider a wireless signal-propagation model with randomly positioned signal sources and reflections from walls, and suppose the model has sufficient randomness and regular walls.…

Matheron's characterization conjecture. Matheron's inequalities are not only necessary but also sufficient to characterize a first-order stationary indicator variogram.

Triple-sum normal approximation conjecture. Under these modified locality and stabilization definitions, it should be possible to control the difference operators associated with t…

Let $bb_c(L)$ be the critical intensity for the Poisson stick model in hyperbolic space $bbH^d$, and let $L$ denote the stick length. Higher-dimensional hyperbolic conjecture. Some…

Let $bb^E_c(L)$ and $bb^E_u(L)$ denote the critical and uniqueness values for the Poisson stick process in $bb^d$, defined analogously to the corresponding hyperbolic quantities. E…

Let $bb^E_c(L)$ and $bb^E_u(L)$ denote the critical and uniqueness values for the Poisson stick process in $bb^d$, defined analogously to the corresponding hyperbolic quantities. E…

Chenavier–Robert's conjecture. The extremal index is conjectured to be

Let $A$ be a compact Riemannian manifold-with-boundary of dimension $d\geq 2$ and Riemannian volume $v(A)\in(0,\infty)$. The paper's coverage quantities and theorems are originally…

Let $\mathcal V_{\text{typ}}$ be the typical cell of a Poisson–Voronoi tessellation in dimension $d$, with intensity $\lambda=\lambda(d)>0$, and let $B$ denote the unit ball in…

Let $d\geq 2$. For $d+2$ independent random points in $\mathbb{R}^d$ drawn from the beta distribution with parameter $\beta>-1$, let $p_d(\beta)$ denote the probability that the po…

Torus extension conjecture. It should be possible to extend the Poisson approximation results of Theorems 1–3 to $mathbb T^d$.

Let $F^{(m)}_r$ denote the $m$-th intersection functional of the Poisson $k$-flat process in hyperbolic $d$-space, with $2k>d+1$. A standard central limit theorem means convergence…

Kendall's conjecture. Almost surely, in the planar case $d=2$, the speed of every geodesic is bounded away from zero away from its endpoints; equivalently, the non-trivial road seg…