Stability of Minkowski inequality for nearly spherical sets
Abstract
In this paper, we study the stability of Minkowski inequality for nearly spherical domains that are C1 close to the ball. We show the stability inequalities between the positive part of the σk curvature integrals for C1 perturbations of a ball; we also establish the stability inequalities for axially symmetric C1 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.
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