The min-max theory for the area functional is a Morse theory on the space of surfaces contained in a three-dimensional Riemannian manifold. The theory experienced remarkable developments and found deep applications in differential geometry. The min-max widths are invariants that naturally emerge from this theory as special critical values of the area. It is very interesting to compare these numbers to other geometric quantities, such as the volume and curvature bounds of the ambient manifold. 

In this talk, we briefly discuss some classification results of Riemannian manifolds involving the min-max widths of the area functional. The main topic of this talk is a new notion of width of circles embedded in Riemannian manifolds, generalizing the classical one for plane curves. When this width coincides with the diameter of the curve with respect to the distance function of the ambient space, the curve has properties similar to those of plane curves of constant width. We also found a connection between curves whose width equals half of their length and Riemannian fillings of the circle.