Quantitative Stability for Minkowski's problem

We derive quantitative stability results for Minkowski bodies, as well as their counterparts, the $L_p$-Minkowski bodies in the range $1 \le p \neq n$. We prove that, for every pair of probability measures $\mu,\nu$ satisfying a quantitative form of the classical dispersion assumptions yielding existence of such bodies, we have a control of the form $$\inf_{x\in \mathbb{R}^n}\mathrm{d_H}(E_\mu, x + E_\nu) \le C \mathrm{d_C}(\mu,\nu)^{\frac{1}{n-1}}, \quad \alpha(E_\mu, E_\nu)^2 \le C \mathrm{d_C}(\mu,\nu)^{1 + \frac{1}{n-1}},$$ where $\mathrm{d_H}$ denotes the Hausdorff distance, $\alpha$ denotes the Fraenkel asymmetry and $\mathrm{d_C}$ is the dual-convex distance of probability measures on the sphere. Our arguments are based on a variational problem whose optimizers are Minkowski bodies, for which we can obtain strong-concavity properties with the quantitative Brunn-Minkowski and isoperimetric inequalities. While the exponent in the Hausdorff distance is sharp, the exponent in the Fraenkel asymmetry is optimal in dimension $2$.