Appears in collection : Not Only Scalar Curvature Seminar

In his Four Lectures, Gromov formulated a conjecture regarding the scalar curvature extremality property of convex polytopes. Recently, assuming the matching angle hypothesis, S. Brendle provided a proof using Dirac operator techniques along with a smoothing construction. Additionally, Gromov outlined a proof of this conjecture, specifically addressing cases with acute dihedral angles. In this presentation, I will provide a brief summary of recent developments in the dihedral rigidity problem. I will also discuss joint work with S. Brendle, where we introduce an alternative smoothing construction for Gromov's argument. Our proof of the rigidity statement relies on a deep estimate due to Fefferman and Phong.