Stability of Minkowski inequality for nearly spherical sets

In this paper, we study the stability of Minkowski inequality for nearly spherical domains that are $C^1$ close to the ball. We show the stability inequalities between the positive part of the $\sigma_k$ curvature integrals for $C^1$ perturbations of a ball; we also establish the stability inequalities for axially symmetric $C^1$ perturbations of a ball. Finally, we construct a counterexample, illustrating that the inequalities become invalid if we do not compensate the integral with the negative part of the curvature. Our work generalizes Glaudo's results on the mean curvature integral to the fully nonlinear cases.