Allan Wechsler acwacw at gmail.com I think that in either model of hyperbolic space, any nontrivial polyhedron with its vertices on the outer sphere has infinite volume.

--No! Always has finite volume! To solve in 2 dimensions, the answer is a regular N-gon. The hyperbolic area of an N-gon with all vertices "at infinity" is the same as the area of N-2 triangles with vertices at infinity (glue them together to make N-gon) and the area of a triangle with all angles=0 is pi (since angle deficit = area). Hence maximal hyperbolic area of N-gon is (N-2)*pi. In D dimensional hyperbolic space with N=D+1 answer is always a regular simplex (a result proven by Haagerup & somebody). Albeit note when D=2 every triangle at infinity is congruent to every other so the word "regular" has no meaning in that case. But when D=3, not every tetrahedron with vertices at infinity is congruent to every other... It is quite interesting to find out the maxvol polytopes in hyperbolic D-space... and the volumes are always finite. -- Warren D. Smith http://RangeVoting.org <-- add your endorsement (by clicking "endorse" as 1st step)