Hyperbolic Simplices of Maximal Inradius

For $n\in \mathbb{N}$, consider a hyperbolic $n$-dimensional simplex $\Delta$, defined by $1+n$ points in the compactified hyperbolic space $\mathbf{H}^n \sqcup \partial \mathbf{H}^n$. For each integer $m\le n$, denote $\delta^n_m(\Delta)\in [0,+\infty]$ the Hausdorff distance between its skeleta of dimensions $n$ and $m$. In particular, $\delta^n_{n-1}(\Delta)$ is its inradius. The maximum of $\delta^n_m(\Delta)$ over $\Delta\in (\mathbf{H}^n \sqcup \partial \mathbf{H}^n)^{1+n}$ is denoted $\mu^n_m\in [0,+\infty]$. We first show that $\Delta$ has maximal inradius $\delta^n_{n-1}(\Delta)=\mu^n_m$ if and only if its is (total) ideal and regular; for which the inradius is given by $\tanh \mu^n_{n-1} = 1/n$. We deduce that $\Delta$ has maximal $\delta^n_{n-1}(\Delta)=\mu^n_m$ if and only if it is (total) ideal and regular. We compute that the maximal distance to the $1$-skeleton $\mu^n_1$ is given by $\left(\tanh \mu^n_1\right)^2 = (n-1)/(2n)$ and deduce that those are uniformly bounded by $\lim_{n} \mu^n_1 = \log(1+\sqrt{2})$.