Costin Vîlcu
Simion Stoilow Institute of Mathematics of the Romanian Academy, Bucharest, Romania

This talk is based on a joint work with Joseph O'Rourke (Smith College, USA).

Given a convex polyhedral surface $P$, we define a tailoring as excising from $P$ a simple polygonal domain that contains one vertex $v$, and whose boundary can be sutured closed to form a new convex polyhedron via Alexandrov's Gluing Theorem. In particular, a digon-tailoring cuts off from $P$ a digon containing $v$, a subset of $P$ bounded by two equal-length geodesic segments that share endpoints, and can then zip closed.

In the first part of this talk, I will present properties of the tailoring operation on convex polyhedra. The main result is that $P$ can be reshaped to any polyhedral convex surface $Q$ inside $P$ by a sequence of tailorings.

In the second part of this talk, I will present vertex-merging processes on convex polyhedra (each vertex-merge being in a sense the reverse of a digon-tailoring), creating embeddings of $P$ into enlarged surfaces. The aim is to produce non-overlapping polyhedral and planar unfoldings.