In joint work with Costin Vilcu, we prove that every convex polyhedron P can be reshaped to any other convex polyhedron Q inside P via a sequence of a polynomial number of "digon tailorings." A digon is a subset of (the surface of) P bounded by two equal-length geodesic segments that share endpoints x and y. A digon tailoring step excises a digon that contains a single vertex v, and then sutures closed the two sides of the digon. Informally, if the surface were made of paper, one could view digon tailoring as pinching a neighborhood of v flat, slicing off v, and then identifying the two sides of the slice.
Corollaries include enlarging Q to P and continuously folding P onto Q.