A renowned conjecture—described by A. Ros as one of the nicest open problems in classical geometry—proposes that the sets minimizing the perimeter among subsets of the 3-dimensional torus R^3/Z^3 with a fixed volume v are spheres, cylinders, or stripes (depending on the value of v). An equivalent conjecture applies to isoperimetric sets in the 3-dimensional cube (0,1)^3.
In this talk, we will first introduce the isoperimetric problem as a quintessential geometric variational problem, and then focus on some recent progress related to the above-mentioned cube conjecture.
Part of the talk will discuss an ongoing project with G. Antonelli. It is aimed at a non-expert audience.
