Hyperbolic Simplices of Maximal Inradius

Abstract

For $n\in \mathbb{N}$, consider a hyperbolic $n$-dimensional simplex $\hat{I}”$, defined by $1+n$ points in the compactified hyperbolic space ${\mathbf{H}}^n\bigsqcup \partial {\mathbf{H}}^n$. For each integer $m\le n$, denote $\hat{I}{´}_m^n(\hat{I}”)\in [0,+\infty ]$ the Hausdorff distance between its skeleta of dimensions $n$ and $m$. In particular, $\hat{I}{´}_{n-1}^n(\hat{I}”)$ is its inradius. The maximum of $\hat{I}{´}_m^n(\hat{I}”)$ over $\hat{I}”\in ({\mathbf{H}}^n\bigsqcup \partial {\mathbf{H}}^n{)}^{1+n}$ is denoted $\hat{I}{¼}_m^n\in [0,+\infty ]$. We first show that $\hat{I}”$ has maximal inradius $\hat{I}{´}_{n-1}^n(\hat{I}”)=\hat{I}{¼}_m^n$ if and only if its is (total) ideal and regular; for which the inradius is given by $\tanh \hat{I}{¼}_{n-1}^n=1/n$. We deduce that $\hat{I}”$ has maximal $\hat{I}{´}_{n-1}^n(\hat{I}”)=\hat{I}{¼}_m^n$ if and only if it is (total) ideal and regular. We compute that the maximal distance to the $1$-skeleton $\hat{I}{¼}_1^n$ is given by ${\left(\tanh \hat{I}{¼}_1^n\right)}^2=(n-1)/(2n)$ and deduce that those are uniformly bounded by ${\lim }_n\hat{I}{¼}_1^n=\log (1+\sqrt{2})$.

14 pages, 26 figures