The ?-Delaunay tessellation III: Kendall’s problem and limit theorems in high dimensions

The $\beta$-Delaunay tessellation in $\mathbb{R}^{d-1}$ is a generalization of the classical Poisson-Delaunay tessellation. As first result this paper we show that shape weighted typical cell tessellation, conditioned on having large volume, close to regular simplex $\mathbb{R}^{d-1}$. This generalizes earlier results Hug and Schneider about (non-weighted) simplex. Second, asymptotic behaviour volume cells high-dimensional analysed, as $d\to\infty$. In particular, various high dimensional limit theorems, such quantitative central theorems well moderate deviation principles, are derived.