On the maximal volume of three-dimensional hyperbolic complete   orthoschemes
  A three-dimensional orthoscheme is defined as a tetrahedron whose base is a right-angled triangle and an edge joining the apex and a non-right-angled vertex is perpendicular to the base. A generalization, called complete orthoschemes, of orthoschemes is known in hyperbolic geometry. Roughly speaking, complete orthoschemes consist of three kinds of polyhedra; either compact, ideal or truncated. We consider a particular family of hyperbolic complete orthoschemes, which share the same base. They are parametrized by the "height", which represents how far the apex is from the base. We prove that the volume attains maximal when the apex is ultraideal in the sense of hyperbolic geometry, and that such a complete orthoscheme is unique in the family. 
Kazuhiro Ichihara and Akira Ushijima
math.MG math.GT