Can you have a non-convex polyhedron with bigger volume than a convex
polyhedron with isometric surface? Imagine you start blowing air into a
polyhedron with a bendable but non-stretchable surface. What can be
said about the limiting shape? These are the key questions I will
consider. I will start the talk by discussing the history of the
problem, presenting examples, and surveying earlier work on realization
of surfaces. In the second part of the talk I will discuss my recent
results on volume-increasing deformations of polyhedral surfaces.