20 problems

Let $n,d\geq 1$, and let $x_1,\ldots,x_n$ be independent random vectors with … For an ellipsoid $\mathcal{E}$, say that it is an ellipsoid fit to $(x_i)_{i=1}^n$ if every point…

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…

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 $\mathcal V_{\text{typ}}$ 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 $K$ be a regular $n$-sided polygon with $n\geq 3$, circumscribed by a circle of unit radius. For a positive even integer $p$, let $L_{22}^{(p)}$ denote the corresponding…

Let $U_1,\ldots,U_N$ be independent random vectors in $\mathbb{R}^d$ with beta density … where $\beta>-1$, $\mathbb B^d=\\{x\in\mathbb{R}^d:\\|x\\|\leq 1\\}$, and $v_m$ denotes the…

Let $d\in\mathbb{N}$ and $k\in\{0,\ldots,d-1\}$. Write $\mathcal{V}_d$ for the Poisson–Voronoi polytope in dimension $d$, and let $\mathbb{E}f_k(\mathcal{V}_d)$ denote its expected…

Let $n\in\mathbb{N}$ and $k\in\{1,\ldots,n\}$. Let $\beta>(n-1)/2$ be an integer or half-integer, and let $\tilde{\mathbb{J}}_{n,k}(\beta)$ denote the corresponding dual expected i…

Let $\beta\geq -1$ be an integer or half-integer, and let $n\in\mathbb{N}$ and $k\in\{1,\ldots,n\}$. The quantity $\mathbb{J}_{n,k}(\beta)$ denotes the expected internal-angle sum…

Curvature coefficient conjecture. The universal constant is

Let $X^\lambda$ be a homogeneous Poisson point process on $\mathbb{R}^2$, so that $\Lambda\equiv |\cdot|$, and let $g_{(\gamma,N_0,\tau)}(X^\lambda)$ be the SINR graph with noise p…

Let $t$ be a time, let $\Xi_{\le t}$ be the set of particles arriving by time $t$, and let $A(t,\xi)$ denote the measurable region associated with $\xi\in\Xi_{\le t}$. The regions…

Let $K$ be a convex subset of ${\mathbb R}^d$, let $u$ be a unit vector in ${\mathbb R}^d$, and let $\mathsf{S}_uK$ denote the Steiner symmetrization of $K$ along $u$. For…

Let $\phi_0$ be the point process on $\mathbf{S}$ corresponding to the stationary distribution of $\phi_t$, with intensity $\beta$. Let $\psi$ be an independently marked Poisson Po…

Let $L_n$ be the random polygon sequence generated by the consecutive random subdivision, and suppose that $u$ satisfies … and that the support of $\xi$ contains at least two dist…

Let $m$ be the number of unbounded trees, and condition on $m\in\{2,3,4,5\}$. For cyclically labeled interfaces, define the sector widths … with the conventions…

Let $\Pi_0^\mathrm{s}$ be the vertices of the square lattice with side length $1$, and let $\Pi_\epsilon^\mathrm{s}$ be obtained by moving those nodes independently according to Br…

Let $\Pi_0$ be the hexagonal circle-packing point process, let $\Pi_t$ be obtained by moving its nodes independently according to Brownian motion for time $t$, and let $R(\Pi_t)$ b…