On Luxemburg’s problem

In 1967, W.A.J. Luxemburg raised a problem: Determine all the extreme points of the set of elements majorised by an integrable function on an arbitrary finite measure space. The case of atomises measure space has been resolved by Ryff. However, the general case remained elusive. During the past decades, several authors have also worked on the non-commutative version of Ryff’s result. We provide a complete resolution of Luxemburg's problem, which unifies Ryff’s result and the classic result for vectors with significant extension. Moreover, we obtain a noncommutative version of this result. This joint work (Dauitbek, Huang, S.) is published in Advances in Mathematics 2020.