Hyperbolic Simplices of Maximal Inradius

TL;DR

This paper characterizes hyperbolic simplices with maximal inradius and other maximal distances, showing they are ideal and regular, and provides explicit formulas and bounds for these maximal distances in hyperbolic space.

Contribution

It proves that maximal inradius and other maximal distances occur only in ideal, regular simplices and computes explicit formulas and bounds for these maximal distances.

Findings

Maximal inradius occurs in ideal, regular simplices with inradius anh^{-1}(1/n).

Maximal distance to the 1-skeleton is anh^{-1}(rac{ ext{sqrt}((n-1)/(2n))}).

Maximal distances are uniformly bounded, approaching ext{log}(1+ ext{sqrt}(2)) as n o fty.

Abstract

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 $\lim_{n} \mu^n_1 = \log(1+\sqrt{2})$.