13 problems

Let $N,dfin\mathbb{N}$ satisfy $N\geq d+2$. For a partition $[N]=A\cup B$, let $\mathbb{P}_d(A,B)$ denote the probability that $(A,B)$ is a Radon partition among $N$ independent ra…

Let $d$ and $n$ be as in the beta-polytope model, and fix parameters $\beta_1,\ldots,\beta_n\geq -1$. Write $\mathscr{P}_{n,d}^{\beta_1,\ldots,\beta_n}$ for the corresponding beta…

For $n\in\mathbb{N}$, let $N_n$ be a Poisson random variable with parameter $n$, and let $\mathcal{P}_{N_n,d}^{\bbbeta}$ and $\mathcal{P}_{n,d}^{\bbbeta}$ denote the corresponding…

Let $\bbbeta=(\beta_1,\ldots,\beta_m)$ be the block-parameter vector for the block-beta random polytope $\mathcal{P}_{n,d}^{\bbbeta}$. Negative-parameter extension conjecture. The…

Let $d=d_1+\cdots+d_m$, let $\bbbeta$ be the block-parameter vector, and let $\mathcal{P}_{d,d}^{\bbbeta}$ be the associated block-beta random polytope. All-face-dimensions conject…

Let $m\in\mathbb{N}$. For $i\in\\{1,\dotsc,m\\}$, let $K_i\subset \mathbb{R}^{d_i}$ be a convex body whose boundary is twice differentiable with positive Gaussian curvature everywh…

Let $P$ be a simplicial $d$-polytope with $2d$ vertices. Two facets are estranged if they are disjoint. Von Stengel's conjecture. The maximum number of pairs of estranged facets of…

Let $\mathcal Z_d^{\text{typ}}$ denote the typical Poisson zero cell, and let $f_0(\mathcal Z_d^{\text{typ}})$ be its number of vertices. Large-deviation conjecture. As…

Fix $k\in\{2,3,\ldots\}$ and sample $k$ vertices uniformly at random from the vertices of the random polytope $\operatorname{conv}\Pi_{d,1}$. Let…

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…