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 $E \subset \mathbb{R}^n$ with $|E| = |B|$ and a barycenter at the origin, one has $P(E) - P(B) \ge c(n)|E\Delta B|^2$. 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.