Quantitative quermassintegral inequalities for nearly spherical sets

In this paper, we establish quantitative Alexandrov–Fenchel inequalities for quermassintegrals on nearly spherical sets. In particular, we bound the [Formula: see text]-isoperimetric deficit from below by the Fraenkel asymmetry. We also find a lower bound on the [Formula: see text]-isoperimetric deficit using the spherical deviation.