43 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'…

Let $K$ be a smooth convex body with volume one in ${\mathbb R}^d$. Let $\Pi_n$ be the Poisson random polytope in $K$ and let $K_n$ be the convex hull of $n$ independent uniformly…

Let $K$ be a convex set with volume one in ${\mathbb R}^d$. Choose $n$ random points independently and uniformly in $K$, and let $K_n$ be their convex hull. Write…

Symmetric maximizer conjecture. The symmetric convex bodies maximizing the relevant quantities $\mathbb{E}U_{K,N}^p$ and $\mathbb{E}V_{K,N}^p$ should be either parallelotopes or cr…

Simplex conjecture. For every $n\geq 2$, $N\geq n+1$, and $p\geq 1$, the quantity $\mathbb{E}U_{K,N}^p$ achieves its maximum exactly when $K$ is a simplex.

Let $B_\infty^n$ denote the unit ball of $\ell_\infty^n$, and for $K\in\mathcal{K}_s^n$ let $U_K=U_{K,n+1}$ and $V_K=V_{K,n}$ be the normalized volumes of a random simplex and rand…

Let $\mathcal{K}^n$ be the family of convex bodies in $\mathbb{R}^n$, let $\mathcal{K}_s^n$ be the family of symmetric convex bodies, and let $U_{K,N}$ and $V_{K,N}$ denote the nor…

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 $V_s(\widetilde K_n^\beta)$…

The preceding variance bound concerns triangulations in the regime where $p$ may approach $1$. Asymptotic equality conjecture. The result should hold with asymptotic equality, and…

Let $K$ be a convex body and $L$ a convex body defining the $L$-convex floating body and wet part. In the planar case with $L=RB^2$, suppose that the Bárány–Larman-type relation fo…

Random-polytope expansion conjecture. There exists a constant $c>1$ such that, for all $p\leq 0.999$ with $p2^n\rightarrow\infty$, with high probability,

Random-polytope expansion question. Is it true that, for $p=o(1)$ and $p2^n\rightarrow\infty$, with high probability,

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…

Volume-ratio asymptotic conjecture. In this regime,

Let $d$ be a positive integer and let $K$ be a $d$-dimensional convex body of volume $1$. A hyperplane section of $K$ is a $(d-1)$-dimensional intersection obtained by cutting $K$…

Let $K\subseteq\mathbb{R}^n$ be a symmetric convex body, and let $V_{N,K}$ denote the volume of the random polytope obtained by choosing $N$ points together with their antipodal po…

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 $\mathcal C(n,M)$ be the set of combinatorial types of simple $n$-polytopes with $M$ faces. Assign equal probabilities to all elements of $\mathcal C(n,M)$, and let $X$ be a ra…

Let $\Sigma=\{\sigma_i\}_{i=1}^{N^{n-1}}\subset S^{n-1}$ be randomly chosen points on the unit sphere, and let $X_N=X(\Sigma)$ be the convex polyhedron defined by the tangent hyper…

Let $d\geq 1$ and let $H_1,\ldots,H_j$ be hyperplanes through the origin of $\mathbb{R}^{d+1}$, otherwise in general position, with $j\in\{1,\ldots,d\}$. Define … Let…