The paper provides optimal quantitative stability estimates for the celebrated Alexandrov’s Soap Bubble Theorem within the class of \(C^{k,\alpha }\) domains, for any \(k \ge 1\) and \(0 < \alpha \le 1\) , by leveraging Gagliardo-Nirenberg-type interpolation inequalities. Optimal estimates of uniform closeness to a ball are established for \(L^r\) deviations of the mean curvature from being constant, for any \(r\ge 2\) (more generally, for any \(r>1\) such that \(r\ge (2N-2)/(N+1)\) ). For \(r>\frac{N-1}{2}\) , the stability profile is linear, thus returning the existing results established in the literature through computations for nearly spherical sets. All the stability estimates for \(r\le \frac{N-1}{2}\) , for which the profile is not linear, are new; even in the particular case \(r=2\) (which has been extensively studied, since it is a case of interest for several critical applications), the sharp stability profile that we obtain is new. Interestingly, we also prove that the (non-linear) profile for \(r \le \frac{N-1}{2}\) improves as k becomes larger to such an extent that it becomes formally linear as k goes to \(\infty \) . Finally, for any \(k \ge 1\) and \(0< \alpha \le 1\) , we show that our estimates are optimal within the class of \(C^{k,\alpha }\) domains, by providing explicit examples.
