Min-max theories for the area functional have undergone ground-break developments in recent years. One aspect of these theories is that they define many notions of "width" that can be understood as geometric invariants of compact Riemannian manifolds. As such, it is an interesting question to understand what sort of geometric information they encode. In this talk, we will focus on the Simon-Smith width of Riemannian three-dimensional spheres, discussing how large it can be among metrics normalised to have the same volume, and necessary conditions for maxima under further restrictions. This is a joint work with Rafael Montezuma (UM-Amherst).