The isoperimetric problem for hyperideal polyhedra, part I

The isoperimetric problem for hyperbolic hyperideal polyhedra is studied. A convex hyperideal polyhedron, with all vertices beyond the sphere at infinity and equal edge lengths, uniquely maximizes volume among those with the same combinatorial type and surface area. Constructed examples include hyperideal counterparts of all convex uniform polyhedra, certain Johnson solids, and other hyperideal polyhedra with no analogues in Euclidean space.