For a hyperbolic simplex \(T\), some of whose vertices lie outside the hyperbolic space, the intersection of \(T\) with the half-spaces polar to some of the ultra-ideal vertices is called a hyperbolic truncated simplex. This class of polytopes includes many interesting members such as simplices, rectangular hexagons, Lambert cubes, straight simplicial prisms and, most notably, provides a way to construct hyperbolic Coxeter polytopes, in particular those giving rise to “smallest” hyperbolic orbifolds in various senses (see e.g. [R. Kellerhals, Comput. Methods Funct. Theory 14, No. 2–3, 465–481 (2014; Zbl 1307.57001)]).
The author computes the radius of a maximal sphere contained in an arbitrary truncated simplex (see Section 3.2). In Section 4.2, he applies this result to numerically compute the answer for some Coxeter polytopes that give rise to “small” hyperbolic orbifolds. This computation is motivated by the observation that the inradii of such small Coxeter polytopes are related to hyperbolic ball packings of large density.