In the field of geometric inequality stability, it is common to observe a lower bound for the energy difference that grows with a power of 2. For instance, a notable finding by Fusco, Maggi, and Pratelli says that, for any set of finite perimeter E⊂Rn with |E|=|B| and a barycenter at the origin, one has P(E)−P(B)≥c(n)|E∆B|. This phenomenon also appears in some other follow-up work. This pattern is also evident in subsequent research. In my presentation, I will discuss recent findings concerning scenarios in Euclidean spaces where the power is no longer 2 in Euclidean spaces.
