3 problems

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 $\Pi_{d,\alpha}$ be the random point set whose convex hull is the polytope under consideration. A polytope is $k$-neighborly if the convex hull of every $k$ vertices is a…