On volumes of truncated tetrahedra with constrained edge lengths (Q2322533)

Let \(P\) be a tetrahedron and let \(P^*\) be the combinatorial polyhedron obtained by removing from \(P\) small open stars of the vertices. We call lateral hexagon and truncation triangle the intersection of \(P^*\) respectively with a face and with the link of a vertex of \(P\). A truncated tetrahedron is a realization of \(P^*\) as a compact polyhedron \(\Delta \subseteq \mathbb{H}^3\) in hyperbolic space, such that the truncation triangles are geodesic triangles, the lateral hexagons are geodesic hexagons, and truncation triangles and lateral hexagons lie at right angles to each other.\par For \(l > 0\) denote by \(\mathcal{T}_l\) the set of isometry classes of truncated tetrahedra whose internal edge lengths are not smaller than \(l\). We also denote by \(\Delta_l \in \mathcal{T}_l\) the (isometry class of the) regular truncated tetrahedron with all edge lengths equal to \(l\). Set \(l_0 = \cosh ^{-1} \big(\frac{3 + \sqrt{3}}{4}\big)\). The main result of this paper shows that, for every \(l \leq l_0\), the tetrahedron \(\Delta_l\) is the unique element of \(\mathcal{T}_l\) having the maximal volume.