[ambkbuev] Random polyhedron

Consider a polyhedron defined by a set of planes (half spaces).  Is its volume infinite?  If not, which vertex is farthest from the origin, assuming the origin is enclosed?  Quadratic programming can solve this problem, though one wonders if there is anything easier.

The original motivation was to create a cut-gemstone-like random convex polyhedron to demonstrate ray tracing.

Given a vertex, what are its neighboring edges and faces?

Given one of the defining planes, is it redundant, that is, does it contribute to any face?

The above question can be avoided if all planes are tangent to the same sphere, which creates a sphere-like polyhedron. Generalizing, consider a polyhedron defined by a set of planes tangent to some given "inner" surface, perhaps a subdivision surface.  What vertex is furthest from the inner given surface?

Meta: why are questions in computational geometry difficult?